\section{Introduction}
The theoretical prediction \cite{1,2,3,4,5,6} that the combination
of spin-orbit coupling, Zeeman spin splitting, and ordinary
$s$-wave superconductivity could lead to an effective topological
superconducting phase under appropriate (and experimentally
achievable) conditions has led to an explosion of theoretical
and experimental activities \cite{6} in semiconductor nanowires
(InSb or InAs) in proximity to a superconductor (NbTi or Al)
in the presence of an external magnetic field.
The experimental finding \cite{7} of a ZBP, in precise agreement with the
theoretical predictions in the differential tunneling conductance of an
InSb nanowire (in contact with a NbTiN superconducting substrate) at
a finite external magnetic field ($B\sim 0.1-1$ T), followed by independent
corroborative observation \cite{8,9,10,11} of such ZBP both in InSb and InAs
nanowires in contact with superconducting Nb and Al by several groups,
has created excitement in the condensed matter physics community as well as
the broader scientific community as perhaps the first direct evidence supporting
the existence of the exotic, the elusive, and the emergent unparied Majorana bound state
in solids. Such excitement has invariably been followed by a wave of skepticism as one
would expect in a healthy and active scientific discipline with questions
ranging all the way from whether such ZBP could arise from other (i.e. non-Majorana)
origin to whether all aspects of the observed experimental phenomenology
are consistent with the putative theoretical predictions on the topological
superconductivity underlying the existence of the Majorana mode.

One particular issue, which is also the subject of the current work,
attracting a great deal of theoretical attention \cite{12,13,14,15,16,17,18,19,20,21,22,25,26}
is the role of disorder in the Majorana physics of superconductor-semiconductor hybrid
structures. Disorder plays a key role in the Majorana physics because the underlying
topological superconducting phase hosting the Majorana mode (at defect sites)
is essentially an effective spinless p-wave superconductor \cite{1,2,3,4,5},
which, unlike its $s$-wave counterpart, is not immune to non-magnetic elastic disorder
(i.e. spin-independent momentum scattering) as was already known ten years ago \cite{23}.
Thus, even the simplest kind of disorder, namely zero-range random non-magnetic point
elastic scatterers in the wire, could strongly affect the topological superconductivity
in contrast to ordinary $s$-wave superconductivity which is immune to non-magnetic
disorder, and the associated Majorana bound states by suppressing the (topological)
superconducting gap \cite{13} and/or creating Andreev bound states in the superconducting
gap near zero-energy \cite{14} complicating the observation and the interpretation
of the ZBP. We do, however, mention that elastic disorder or momentum scattering in the superconductor
itself, no matter how strong (as long as it does not destroy the superconductor),
 does not affect the topological superconducting phase in the semicondutor. \cite{21}
In addition, it has recently been emphasized that elastic disorder by itself
could create a zero-bias peak (essentially, an anti-localization peak associated with the
disorder-induced quantum interference) in the non-topological phase in the presence
of spin-orbit coupling and Zeeman splitting. Since the precise topological quantum
critical point (as a function of the applied magnetic field) separating the topological
and the trivial superconducting phase is in general not known in the experiments, \cite{7,8,9,10,11,12} one cannot
be absolutely sure that the observed ZBP is indeed a Majorana bound state (MBS)
signature in the topological phase and not a trivial anti-localization peak in the non-topological
phase.

In the current work we consider disorder effects on MBS physics by directly calculating the density of states (DOS)
of finite disordered nanowires in the presence of proximity-induced superconductivity taking into account
spin-orbit coupling and spin splitting arising respectively from Rashba and Zeeman effects in the
wire. We use random uncorrelated point scatterers with $\delta$-function potential in the wire to represent
the elastic disorder. The theory follows the standard prescription \cite{24} of an exact diagonalization
of the BdG equations in a minimal tight-binding model including superconducting pairing, Rashba spin-orbit
coupling, and Zeeman splitting in the Hamiltonian. The diagonalization of the
discretized tight-binding Hamiltonian leads to the exact eigenstates of the system, which then
immediately give the density of states (theoretical details are available in Ref.~\onlinecite{24} and are not
repeated here.) Comparison with theory in the presence of disorder neccessitates the ensemble averaging over many different impurity
configurations since each disorder configuration produces its own unique result with random impurity-induced
delta-function peaks in the superconducting gap.

\section{DOS in the disordered topological phase}
In this section, we consider the effect of disorder on the SM/SC system starting from the topological phase in the clean
limit.
In Fig. 1 (with 8 panels, each representing a different strength of disorder keeping all other parameters fixed),
we show our numerical DOS as a function of energy (E) for different relevant parameter sets in the topological phase of the system. We
show both the ensemble averaged DOS using many-impurity configurations(but using the same disorder strength, i.e.
the same impurity density and potential strength, changing only the random localitions of the impurities along the wire)
and the typical DOS for a single impurity configuration in each case. Each panel corresponds to a specific disorder strength
(i.e. a fixed impurity density) and shows results for three different lengths of the
nanowire.

Since our calculations follow either Ref.~\onlinecite{24} for the exact numerical treatment of Ref.~\onlinecite{13} for the SCBA theory, we refer
the reader to those references for the technical details, which are actually pretty standard. \cite{19,20,21,22}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.7\textwidth,angle=270]{Fig1.pdf}
\end{center}
\caption{Disorder averaged density of states in the topological phase for the semiconductor nanowire in a magnetic field with Zeeman splitting
 $V_Z=5\,$K, proximity-induced pairing potential of amplitude  $\Delta_0=3\,$K, Rashba spin-orbit coupling strength $\alpha_R=0.3\, eV-\AA$. For
this choice of parameters the clean quasiparticle-gap $\Delta=1.3\,$K and coherence length $\xi=0.3\,\mu m$. The difference panels (a-h)
correspond to different disorder strengths characterized by $E_s=\hbar/2\tau= 10, 50, 100, 200, 460, 1300, 3300, 7300$ mK.
 The corresponding mean-free paths are $\lambda$ are in the panels and the self-consistent Born gaps $E_{SCB}= 1.2, 1.1, 0.9, 0.8, 0.13, 0, 0, 0$K.}
\label{Fig1}
\end{figure}

The superconductor (SC)-semiconductor (SM) hybrid nanowire structure is characterized by a large number of independent parameters, both for
the actual experimental laboratory systems and for our minimal theoretical model. The minimal set of parameters necessary to describe the system
 are the  proximity-induced SC gap ($\Delta_0$) in the SM, the parent SC pairing potential ($\Delta_s$), the Rashba spin-orbit coupling ($\alpha_R$),
the spin splitting $(V_Z)$, the chemical potential ($\mu$), the SM effective mass $(m)$ which defines the SM tight-binding hopping
parameter, the nanowire length $(L)$, and disorder (which we take to be the uncorrelated random white noise potential  associated with randomly
located $\delta$-function in real space spin-independent scattering centers). There are a few additional physically important parameters
which are, however, not independent parameters of the theory: the coherence length in the nanowire $(\xi)$, the SM mean-free path $(\lambda)$
due to disorder, and the number of occupied subbands (i.e. transverse quantized levels) in the nanowire which we take to be one
throughout assuming the system to be in the one-dimensional limit. (The single sub-band approximation, made entirely for the convenience
of keeping the number of parameters in the model to be tractable, is a nonessential approximation, and our qualitative results should be
completely independent of this approximation.) In addition, there is an independent parameter defining the hopping amplitude across the
SC/SM interface which controls the proximity-induced SC pairing gap $\Delta_0$ in the SM in terms of the parent gap $\Delta_s$
in the SC. Finally, the proximity gap ($\Delta$) in the SM in the presence of spin-orbit coupling and finite Zeeman splitting is reduced
from $\Delta_0$ in a known manner. All the details for modeling the disorder
are given in Ref.~\onlinecite{13} where SCBA was used in contrast to
our exact numerical diagonalization in the current work following
Ref.~\onlinecite{24}. One specific goal of our current work is to compare the analytic and simple SCBA theory \cite{13} with the exact tight-binding numerical
analysis to test the limits of validity and the applicability of the SCBA theory which, being analytic, can be used rather easily.

We choose parameters approximately consistent with the InSb/Nb systems studied
experimentally in \cite{7}. These are : $\Delta_0=3$K, $\alpha_R=0.3 eV-\AA$ (corresponding to an effective spin-orbit coupling strength $m^*\alpha_R^2=2.5 K$). Since out interest is in disorder effects, we focus on a range of magnetic fields with a spin splitting $V_Z\sim 5$K (we vary it in a few cases
only to change the proximity to the topological quantum phase transition point, separating the topological and the trivial SC phase). Given that the condition for
the topological SC phase to be realized in the SM is given by
\cite{2} $V_Z>\Delta^2+\mu^2$, where $\Delta$ is the actual
induced gap in the SM, the system should be in the MBS carrying
topological phase for $\Delta<5$K (since $\mu=0$). Given that
$\Delta<\Delta_0=3$K, with the reduction of $\Delta$ below $\Delta_0$ arising from the existence of $V_Z\neq 0$, our system is deep in the topological
phase for the results shown in Fig.1 since $V_Z(=5K)\gg \Delta(=1.3K)$ and $\mu=0$.

Each panel in Fig. 1 corresponds to a different disorder strength
in the system, characterized by the corresponding level broadening $E_s=\hbar/2\tau$
(where $\tau$ is the scattering time -- $\tau=\infty$ in the absence of
disorder) or equivalently the mean free path $\lambda=v_F\tau$ (where $v_F$
is the fermi velocity), both calculated in the SCBA according to Ref.
\onlinecite{13} for the given disorder in the wire. In each panel
(and for each disorder) we show our DOS numerical results for two distinct
wire lengths $L=1.5\,\mu m$ and $L=3\,\mu m$. In each case, we show both the
ensemble averaged DOS results using an averaging over many $(> 1000)$ random
impurity configurations (keeping $E_s,\lambda$ etc fixed) and the result
for a typical single impurity configuration (the distinct crosses or dots
denoting delta functions for the DOS at the value of energy ).
We emphasize that for an infinitely long wire ($L\gg \xi\approx 0.5\,\mu m$
for our case) the DOS in the absence of disorder will vanish
throughout the gap ($\pm 1.3$ K in our case) with a $\delta$-function peak
at $E=0$ associated with the MBS at the wire edges.

To characterize the disorder strength for the results in Fig. 1 (with panels
(a) to (h) with increasing disorder keeping all other parameters fixed ),
we use SCBA for this problem which was developed by us in Ref. \onlinecite{13}.
The SCBA theory provides us with the SC gap in the topological phase for a
given disorder strength, allowing us to compare our direct (and exact) numerical
calculation in the presence of disorder with the SCBA theory. We show the
calculated SCBA gap in each case in the figure captions for the sake of direct
comparison with the exact results in the figures. (The SCBA theory is obviously
an ensemble averaged theory for the infinite system and does not depend on $L$.)
It is clear from the result of Fig. 1 that the analytical SCBA theory of Ref.
~\onlinecite{13} are in excellent qualitative agreement
with the exact ensemble-averaged numerical results for the DOS even for
$E_s\approx \Delta$, where the topological gap essentially vanishes
(panel (e) in Fig. 1) both according to the SCBA (i.e. $E_{SCB}=0$) and
in our numerical results.
While the disorder averaged DOS calculated by exact diagonalization
does not stricly vanish inside the gap,
the gap calculated with the SCBA can be identified with the peaks in the DOS
at the edges of the gap. The closing of the gap within the SCBA coincides
with the disappearance of the dips in the DOS around zero-energy.
 The DOS peak at $E=0$ associated with the MBS is
continuously suppressed with increasing disorder, but quite amazingly
there is a discernible DOS peak at $E=0$ even for $E_s(=3.3K)\gg \Delta(=1.3K)$
 where $E_{SCB}=0$, and at best the topological superconductivity is
gapless.

The remarkable result, which is quite apparent in our Figs. 1 (e)-(g),
is that the MBS peak of the DOS at $E=0$ is actually very robust to
disorder and survives disorder strength substantially larger than that (typically $E_s\sim \Delta$)
destroying the induced
superconducting gap $\Delta$. Thus, in Figs. 1(e)-(g), although the
SCBA theory and our exact numerical results both show the system to be
gapless with $E_s>\Delta$, the DOS peak at $E=0$ associated with the MBS
persists until $E_s\gg \Delta$ as in Fig. 1(h) where $E_s=7 K(\gg \Delta=1.3 \, K)$.
 It is not only that the MBS feature in the ensemble averaged DOS
survives up to very strong disorder (e.g. the mean free path $\lambda=0.5\,\mu m,0.2\,\mu m$ and
$0.06\,\mu m$ respectively in Figs. 1(e)-(g) which are smaller than the
wire lengths of $L=1.5\,\mu m$ and $3\,\mu m$ used in our numerical work),
the typical DOS for single random impurity configuration also shows peaks at
$E=0$ as can clearly be seen in Figs. 1(e)-(g) [ and as well as in Figs.
1(a)-(d)], but not in Fig. 1(h) where the very large disorder strength
($\lambda=0.02\,\mu m$) suppresses both the ensemble averaged MBS peak as
well as the single configuration peak at $E=0$. The survival of the zero-energy DOS
peak well above the point where the SC gap is completely suppressed by disorder
is an important new result of our exact numerical work directly establishing the
possible theoretical existence of a gapless topological SC phase.

\subsection{Griffiths effects}
The origin of the $E=0$ peak in the DOS in the strongly disordered case
can be related to the Griffiths effect previously considered for the spinless
$p$-wave superconductor \cite{23}. The Griffiths effect in the semiconductor nanowire
at finite $V_Z$ arises from the disorder-induced
 variation of the chemical potential, which can lead to a transition from a topological phase
to a non-topological phase and vice-versa. The variation of the effective chemical potential
can lead to domain-walls between topological and
non-topological regions each of which would support a local zero-energy MBS.
Since each region is of finite extent, the MBSs are a finite distance apart and
split into  conventional states with a non-zero energy. In fact, by carefully
considering the distribution of the distances between the MBSs, it has been shown \cite{23}
that this splitting typically leads to a singular peak in the DOS at $E=0$,
which diverges as $E\rightarrow 0$ as a power-law in $E$. The topological phase is of course
characterized by true zero-energy edge states, whose energy is exponentially small in the
length of the system $L$. Thus, in the long wire limit, the topological phase and the non-topological
phase both have power-law divergent DOS due to the Griffiths effects, but the topological phase
has a pair of zero-modes exponentially close to zero energy \cite{12}. This distinction
between exponential versus power-law in the length seems to appear in the DOS plotted in Fig 1.
At small disorder, there is a sharp peak which is very weakly split. At larger disorder (panels (e,f,g,h))
, which is approximately when the gap vanishes i.e. $E_{SCBA}\rightarrow 0$, the $E\sim 0$ peak in the
DOS is broadened.

For finite length systems, the sharp transition between the
topological and non-topological phase at finite disorder becomes a crossover.
While the topological invariant can be computed even for a disordered, but strictly infinite system \cite{12,25,26},
we have not done so in the present work because we restrict ourselves to systems
of wire lengths comparable to the experimental systems. For such finite wires the distinction between the topological
and non-topological phase is not sharp. In fact, the zero-modes arising from the Griffiths effect
are themselves MBSs. The MBSs characterizing the topological phase are special only in the
sense that they occur near the ends of a finite system and are therefore separated from other low-energy
MBSs by a distance of the order of the length of the wire. This can also occur from the Griffiths
effect in some realizations of disorder, even in the non-topological parameter regime.
Therefore, the Griffiths singularities seen in the disordered Fig. 1 indicate the
 presence of several low-lying MBSs in the spectrum of the finite wire.
For very large disorder the system becomes completely non-topological (and gapless)
 as in Fig. 1(h).

There are three important messages following from our DOS results for the
topological phase in the presence of disorder: (1) the zero-energy DOS
peak associated with the MBS is strongly suppressed by disorder; (2) SCBA
is an excellent quantitative approximation for calculating the SC gap in the
topological phase including effects of disorder; and (3) most importantly,
the zero-enenrgy MBS peak is very robust against disorder and survives well
after the SC gap in the topological phase has been suppressed to zero, and
disappears only when the mean free path $\lambda\lesssim \xi/6$ or $E_s\gtrsim 6\Delta$
(for the systems are studied) whereas the topological SC gap vanishes for
$\lambda\lesssim\xi$.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.47\textwidth]{Fig2.pdf}
\end{center}
\caption{Quasiparticle gap $\Delta$ (in the color bar) versus Zeeman potential $V_Z$ and scattering rate $E_s$ in the topological
phase. The black region represents the gapless phase and is therefore not topologically robust.}
\label{Fig2}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.7\textwidth,angle=270]{Fig3.pdf}
\end{center}
\caption{Disorder averaged density of states in the non-topological phase for the semiconductor nanowire in a magnetic field with Zeeman splitting
 $V_Z=0,\,1\,$K (panels $(a,b)$ respectively) at $\mu=0$ for different disorder strengths characterized by scattering rates $E_s=\hbar/\tau$.}
\label{Fig3}
\end{figure}

Given the quantitative validity of the SCBA, we can calculate the phase diagram
of the SC/SM structure in the presence of disorder by using the analytical SCBA
theory. \cite{13} We show our SCBA-calculated quantum phase diagram of the system in
Fig. 2. In Fig. 2, we fix all parameters except for disorder $(E_s)$ and
spin-splitting $(V_Z)$ with the color representing the calculated SC gap
$E_{SCBA}$ in the presence of disorder (by definition $E_{SCBA}=\Delta$
for $E_s=0$). The black region represents gapless superconducivity.
 We note, based on our numerical results of Fig. 1, that a large part of the black
region in the phase diagram allows for well-defined zero-energy DOS MBS peaks
although the system is essentially a gapless topological superconductor in
this regime. Much of this SCBA gapless regime is dominated by Griffiths physics
except for very large disorder when the system eventually becomes nontopological (i.e.
even the finite topological segments disappear).
Much of this SCBA gapless regime is dominated by Griffiths physics except for very large
disorder when the system eventually becomes non-topological.
Except for our current results showing the well-defined robust
persistence of the MBS peak even in the gapless topological regime
(the black region in Fig. 2), one would have concluded that SCBA predicts a rather
gloomy picture for the existence of the Majorana mode in the disordered
SC/SM hybrid structure since, without our current exact results, the
conclusion would have been that there cannot be any MBS in the black region
of the phase diagram. Of course the quantitative details of the SCBA phase diagram
depend on the SO-coupling strength and the topological phase with a large SC gap
is easily achieved by large (small) SO coupling (disorder).

\section{DOS in the disordered non-topological phase}
Next we comment on the effect of disorder in the non-topological
SC regime, i.e., for $V_Z<\sqrt{\Delta^2+\mu^2}$ for the SC/SM hybrid
structure. In Fig. 3(a) and (b), we show our calculated numerical DOS in
the non-topological SC phase (for $\mu=0$) for $V_Z=0$ (3a) and $1K$ (3b) in
the presence of disorder. All parameters other than $V_Z$ are exactly the same
as in Fig. 1. We see that, at least for $\mu=0$, the behavior of the DOS is
similar to a classic $s$-wave SC (even for $V_Z=1\, K$ which simply reduces the
gap from $\Delta_0=3\,K$ to $2\,K$) with essentially no discernible effect
on the DOS. Even for disorder as large as $E_s> 7\,K>2\Delta_0$, we do not see any structure
developing in the SC DOS gap which remains completely flat.
For $V_Z=0$ (Fig. (3a)) this is a direct consequence of Anderson's theorem where the robustness
of the gap arises from time-reversal symmetry. We see in Fig. (3b) that this
behavior persists for small Zeeman fields as long as $V_Z\lesssim \Delta$. This behavior
can be expected based on previous studies on the bound states of single impurities in
spin-orbit coupled nanowires in proximity to superconductors \cite{27}. There it was found that the low-energy
sub-gap states appear for short-ranged impurities only in the topological phase.
However, this conclusion might not apply to longer-range disorder because in principle,
at any non-zero value of Zeeman potential puddles can lead to the formation of a pair of
low-energy MBSs. In our numerical work here we restrict ourselves only to zero-range white noise
disorder in the semiconductor.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.7\textwidth,angle=270]{Fig4.pdf}
\end{center}
\caption{Disorder averaged density of states in the non-topological phase for the semiconductor nanowire in a magnetic field with Zeeman splitting
 $V_Z=4,\,5\,$K (panels $(a,b)$ respectively) at $\mu=5\,$K for different disorder strengths characterized by scattering rates $E_s=\hbar/\tau$.}
\label{Fig4}
\end{figure}

The situation, however, changes qualitatively when we consider the limit $V_Z\gtrsim \Delta$.
 We show these
results in Fig. 4 where $V_Z=1\, K$, $\mu=5\, K$ and $V_Z=\mu=5\,K$ are shown for
several values of the disorder parameter $E_s(\equiv \hbar/2\tau)$. All other
parameters are the same as in Fig. 1 and 2. We note that the situation in Fig. 4
describes the non-topological phase since $V_Z=\mu<\sqrt{\mu^2+\Delta^2}$.
In both panels, large disorder has a strong effect on the SC DOS shrinking the SC gap
considerably. However, in Fig. 4(b), where $V_Z\gg\Delta$,
 eventually for $E_s=0.2\,K$ and $1.3\,K$ (and
$V_Z=\mu=5\,K$), the DOS seems to "flip" and the dip at $E=0$ becomes
a peak at $E=0$ with a concomitant vanishing of any SC gap feature in the
data. For even larger $E_s$ ($E_s=7\,K$ in Fig. 4) the peak feature in the
DOS is suppressed, but there is a clear $E=0$ DOS peak for intermediate
(but still large with $E_s>\Delta$) disorder ($E_s=0.2\,K$ and $1.3\,K$ in
Fig. 4) where the SC gap has disappeared, but a peak has developed in the DOS
at zero-energy. We believe that the zero-energy DOS peak in Fig. 4(b) for
$E_s=0.2\,K$ and $1.3\,K$ has the same origin as the physics recently
discussed in several publications \cite{16,17,18}.
  The hallmark of this "trivial" DOS peak
(arising from the competition among spin splitting, spin-orbit coupling,
and superconductivity) are that (1) it arises only for large Zeeman splitting
 in the non-topological phase; (2) it
occurs only after the SC quasiparticle gap has been completely suppressed by disorder; (3) it
exists only in an intermediate disorder  range where the superconducting quasiparticle gap
has been suppressed, vanishing for larger disorder and the peak becoming a
dip (i.e. the SC gap) for smaller disorder.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.47\textwidth]{Fig5.pdf}
\end{center}
\caption{Quasiparticle gap ($\Delta$) versus Zeeman potential $V_Z$ for different values ($\mu=0,\,5$K) for the chemical potential $\mu$.
The vanishing of the quasiparticle gap marks the topological quantum phase transition from the non-topological phase at small $V_Z$ to the
topological phase at large $V_Z$.
}
\label{Fig5}
\end{figure}

In order to specify where precisely in the phase diagram we are obtaining our
exact numerical results in the presence of disorder we finally show in Fig. 5 our
calculated SC gap as a function of $V_Z$ (for zero disorder) for the
parameters chosen in our calculations. For $\mu=0$ (i.e. Figs. 1-3), the
topological quantum phase transition (TQPT) happens at $V_Z=\Delta_0(=3\,K$
in our choice), and our Fig. 1 belongs to the topological phase ($V_Z=5\,K$, to
the right of the TQPT) whereas our Fig. 2 belongs to the non-topological
($V_Z=0,1\,K$) to the left of the TQPT. Results for Fig. 4 are obtained for
$V_Z=\mu$ (and $V_Z<\mu$) which  never can manifest a TQPT since the
$V_Z=\sqrt{\mu^2+\Delta^2}$ condition cannot be satisfied except for $\Delta=0$
 (i.e. $V_Z\rightarrow\infty$)-- but, even the nontopological SC is strongly
affected by disorder (for $V_Z\gg \Delta$) here since the combination of spin-orbit
coupling and Zeeman splitting makes the Anderson theorem moot.  Thus the DOS peak
in Fig. 4, associated with anti-localization,\cite{16,17,18,19} arises purely in the trivial
phase with a completely suppressed quasiparticle gap which might enable its experimental distinction
from the MBS peak in the topological SC phase.

\section{Discussion}

Finally we mention that the spin-orbit coupled semiconductor system thus has two distinct
topological quantum phase transitions in the presence of the superconducting proximity effect and
Zeeman splitting. The first one is driven by the Zeeman field as originally predicted by Sau et al \cite{2}
with the TQPT defined by $V_Z=\sqrt{\mu^2+\Delta^2}$ assuming a low disorder situation. The second
one is driven by increasing disorder ($E_s$) in the finite Zeeman splitting situation (i.e. $V_Z>\sqrt{\mu^2+\Delta^2}$)
where the topological SC phase is destroyed by disorder (for $E_s\gtrsim \Delta$), leading
to a gapless non-topological phase dominated by the Griffiths physics as originally envisioned by Motrunich et al \cite{23}.
Since the effective SC gap for $V_Z\neq 0$ depends on $V_Z$, and in particular, $\Delta\propto V_Z^{-1}$ for $V_Z\gg \Delta_0$
[see Refs. \onlinecite{5} and \onlinecite{13} for details], we expect that there are two distinct magnetic field driven TQPTs in
the semiconductor nanowires - - the first one is the TQPT predicted in Ref. \onlinecite{2,3,4,5} taking the system from the trivial
$s$-wave SC (induced by proximity effect) to the (effective $p$-wave) topological Majorana carrying
$p-$wave SC phase at low disorder (i.e. $\Delta\gg E_s$), and then the second one at much higher Zeeman splitting
(so that $\Delta\lesssim E_s$) where disorder drives the system from a gapless topological SC to a non-topological
Griffiths phase. It is unlikely that this second (purely disorder driven) TQPT would be experimentally accessibly since the
gapless nature of the SC phase (i.e. black region in Fig. 2 or the situation corresponding to Fig. 1 (e)-(h)) would make the
finite temperature of the experimental system behave like a very high temperature (i.e. $T\gg \Delta$),
making any experimental study of this disorder-driven TQPT difficult, if not impossible. Thus, the effective gapless
nature of the system in Fig. 1(e)-(g) would make it very unlikely that the DOS peak (which is quite obvious in our theoretical
results) could be studied experimentally. The best hope for the direct experimental study of the MBS physics is therefore
to have a large SC gap (as in Fig. 1(a)-(d) or in the non-black region of Fig. 2) in the topological phase which necessitates
having low effective disorder ($E_s\ll \Delta$), a condition guaranteed by having very clean semiconductor wires and or/very strong
SO coupling in the material. We add here that our theoretical results presented in this paper cannot be compared (or connected) with
the experiments in any way since we only calculate bulk DOS of the system, which cannot be directly probed experimentally.

\section{Conclusion}
In summary, we have studied the disorder averaged DOS of a disordered spin-orbit coupled nanowire
in proximity to a superconductor in both the topological and non-topological parameter regime.
The features in the DOS associated with the superconducting gap in the topological phase
appear to be in good quantitative agreement with the SCBA from previous work \cite{13}.
Consistent with previous results \cite{13}, we find that the dips in the DOS
associated with the quasiparticle gap disappear for relatively modest ($E_s\gtrsim \Delta$)
amounts of disorder. However, a zero-energy peak associated with MBSs generated by the Griffiths
effect survives to much higher levels of disorder. The DOS peak associated with different levels of disorder
arising from the Griffiths effect starting from the topological phase and the antilocalization peak starting
from the non-topological phase appear to be different enough that one might distinguish them qualitatively. Of course,
at this point one does not expect a sharp distinction between the peak arising from the Griffith's effect and the
antilocalization effect because they are zero energy peaks in the DOS in the non-topological phase in the same symmetry class i.e. class D.

This work is supported by Microsoft Q, JQI-NSF-PFC, and the Harvard Quantum
Optics Center.


\section{INTRODUCTION}\label{sec:intro}
The TRENDS ({\bf T}a{\bf R}getting b{\bf EN}chmark-objects with {\bf D}oppler {\bf S}pectroscopy) high-contrast imaging program uses precise radial velocity (RV) measurements obtained over a long time baseline to identify promising targets for follow-up observations using adaptive optics (AO) (Crepp et al. 2012b, Crepp et al. 2012c). In addition to diffraction-limited imaging, TRENDS also uses coronagraphy, point-spread function (PSF) subtraction, and $\approx$1 hour integration times to achieve deep contrast at near-infrared wavelengths \citep{marois_06,lafreniere_07,crepp_10}. The primary goal of TRENDS is to image and characterize benchmark M-dwarfs, brown dwarfs, and (ultimately) extrasolar planets, using a combination of complementary substellar companion detection techniques, in order to calibrate theoretical evolutionary models and theoretical spectral models of cool dwarf atmospheres.

Precise Doppler measurements for TRENDS are obtained at Keck Observatory using the HIgh Resolution Echelle Spectrometer (HIRES; \citet{vogt_94}) as part of the California Planet Search program \citep{howard_10}. Targets with the longest RV time baselines have complementary data taken at Lick Observatory using the 0.6m Coude Auxiliary Telescope (CAT) and Shane 3m Telescope with the Hamilton spectrometer \citep{vogt_87}. High-contrast imaging follow-up observations are conducted at Keck using NIRC2 (instrument PI: Keith Matthews) with the Keck II AO system \citep{wizinowich_00}, and at Palomar using Project 1640 with the PALM-3000 high-order AO system \citep{hinkley_11_PASP} .

\citealt{crepp_12b} (hereafter TRENDS I) reported the discovery of three benchmark high-mass ratio binaries, HD~53665, HD~68017, and HD~71881, each companion having an age and metallicity estimate inferred from the solar-type primary \citep{crepp_12b}. Arguably the most interesting object of the three, HD~68017~B, has a RV time baseline of 15 years and shows significant orbit curvature. With a projected separation of only 13.0 AU, it will be possible to measure a precise dynamical mass in the next several years. The model-dependent mass of HD~68017~B from relative photometry is $m_{\rm model}=168\pm21M_J$ ($0.16\pm0.02M_{\odot}$).

\citealt{crepp_13a} (TRENDS II) reported the direct detection of an $m_{\rm model}=0.52\pm0.04M_{\odot}$ tertiary companion orbiting the 84 day period, single-line spectroscopic binary HD~8375~AB. With a separation of $0\farcs3$, HD~8375~C has a brightness comparable to residual scattered starlight from the inner binary pair prior to PSF subtraction. The HD~8375 system represents one of many hierarchical triples in the solar neighborhood that are known to exist, through statistical analysis of multiplicity fractions for stellar systems within $d<10$ pc, but so often evade direct detection \citep{tokovinin_04}.

In this paper (TRENDS III), we report the discovery of a faint object orbiting the nearby, $d=26.14\pm0.37$ pc (\citealt{van_leeuwen_07}), G5 star HD~114174 (LHS~2681, HIP~64150, GJ~9429). The companion, HD~114174~B, cannot be seen without significant on-source integration time and PSF subtraction. It has a projected separation ($0.69\arcsec=18.1$ AU) comparable to the semimajor axis ($a=18.3$ AU) of the brown dwarf HR~7672~B, which recently yielded a dynamical mass with a fractional error of only 4\% \citep{crepp_12a}. Initially characterized as a substellar object based on its brightness and colors, we find that HD~114174~B must instead be a white dwarf, in order to establish a self-consistent interpretation with long-term Doppler RV measurements.

Few ``Sirius-like" systems, i.e., directly imaged white dwarfs around solar-type stars, are presently known -- approximately one dozen within 20 pc \citep{holberg_09}. Our serendipitous discovery of a compact object orbiting HD~114174 thus constitutes a new member of a small collection of nearby, directly detected benchmark white dwarfs for which we can determine physical properties independent of spectro-photometric measurements. Forthcoming measurements will ultimately lead to tight dynamical constraints on its orbit and mass. More than 16 years of RV measurements are already in hand.

\begin{figure*}[!t]
\begin{center}
\includegraphics[height=5in]{f1.eps}
\caption{Relative radial velocity measurements of HD~114174 from the years 1997 through 2013. (Top) A long-term Doppler acceleration of $61.1\pm0.1$ m/s/yr implies the presence of a companion with a long period. (Bottom) A linear fit model results in large residuals indicating systemic curvature (change in the acceleration). Thus, it will soon be possible to calculate the three-dimensional orbit and dynamical mass of HD~114174~B.}
\end{center}\label{fig:rvs}
\end{figure*}

\section{OBSERVATIONS}

\subsection{High-Resolution Stellar Spectroscopy}

\subsubsection{Doppler Measurements}
We obtained RV data of HD~114174 using HIRES at Keck. The standard iodine cell referencing method was used to calibrate instrument drifts and measure precise Doppler shifts \citep{marcy_butler_92}. Our Doppler observations began on January 14, 1997 UT. Several years of measurements revealed that the star exhibits a (mostly) linear acceleration indicative of a distant companion. Fig. 1 shows Doppler observations taken over an 16.0 year timeframe. RV values, epochs, and measurement uncertainties are listed in Table 1.

Treating the Doppler trend as purely linear and using a Markov Chain Monte Carlo (MCMC) fitting procedure, we find an acceleration of $61.1\pm0.1\:\rm{m\:s}^{-1}\rm{yr}^{-1}$. This analysis includes a differential offset between measurements obtained before and after the summer 2004 HIRES detector upgrade, which we treat as a free parameter. Closer inspection of the data shows that the RV slope is slowly changing with time. For instance, the bottom panel of Fig. 1 displays RV residuals once the measurements are subtracted from a linear model. Continued Doppler monitoring will allow us to place tight constraints on the companion orbit and dynamical mass, as soon as the direct imaging astrometry shows curvature. By performing a joint RV-astrometry analysis using presently available data, we derive a lower-limit for the mass of HD~114174~B (Section 5.1) and determine a range of plausible orbital periods (Section 5.4).

\subsubsection{Stellar Properties}
Stellar template spectra, taken with the iodine gas cell removed from the optical path, were analyzed using the LTE spectral synthesis code {\it Spectroscopy Made Easy} (SME) described in \citet{valenti_96} and \citet{valenti_fischer_05}. SME provides an estimate of the stellar effective temperature ($T_{\rm eff}$), surface gravity ($\log g$), metallicity ($\mbox{[Fe/H]}$), and projected rotational velocity ($v \sin i$). The estimated physical properties of HD~114174 derived from spectral fitting are shown in Table 3.

HD~114174 is listed in the SIMBAD database as a G5(IV) subgiant. However, its location with respect to the mean Hipparcos main-sequence \citep{wright_05} and spectroscopic $\log g=4.51\pm0.06$ value indicate the star is a main-sequence dwarf. Placing HD~114174 on a Hertzsprung-Russell diagram, we estimate a mass of $1.05\pm0.05M_{\odot}$ and age of $4.7^{+2.3}_{-2.6}$ Gyr by comparing spectroscopic parameters to Yonsei-Yale theoretical isochrone tracks. Despite its close distance, HD~114174 would rarely, if ever, make typical high-contrast imaging target lists, which nominally only select stars younger than $\approx$1 Gyr \citep{crepp_johnson_11}.

\begin{table}[!ht]
\centerline{
\begin{tabular}{ccc}
\hline
\hline
HJD-2,450,000  &   RV [m~s$^{-1}$]    & Uncertainty [m~s$^{-1}$]   \\
\hline
\hline
     463.1438 &  -403.39   &   1.27 \\
     545.9798  & -390.77   &   1.29 \\
     787.1381  & -351.45   &   1.89 \\
     787.1439  & -337.14   &   1.33 \\
     838.0563  & -333.70   &   1.17 \\
     838.1109  & -329.74   &   1.10 \\
     839.1258  & -334.99   &   1.20 \\
     862.0168  & -319.81   &   1.06 \\
     862.9814  & -320.76   &   1.30 \\
     955.8422  & -313.57   &   1.45 \\
     981.7798  & -311.44   &   1.18 \\
     982.7957  & -312.89   &   1.09 \\
     1009.8359  & -302.12   &   1.13 \\
     1050.7552  & -288.92   &   1.92 \\
     1050.7572  & -290.58   &   1.29 \\
     1171.1521  & -273.81   &   1.35 \\
     1172.1624  & -273.77   &   1.22 \\
     1173.1611  & -277.47    &  1.31 \\
     1227.0404  & -267.37   &   1.28 \\
     1228.0764  & -259.05   &   1.17 \\
     1228.9999  & -262.77   &   1.37 \\
     1310.8726  & -249.58   &   1.33 \\
     1311.8551  & -243.23   &   1.27 \\
     1312.8933  & -247.37    &  1.42 \\
     1313.9960  & -249.02    &  1.40 \\
     1340.8177  & -242.25   &   1.39 \\
     1341.8676  & -248.12   &   1.49 \\
     1342.8364  & -242.09   &   1.21 \\
     1553.1269  & -205.71   &   1.55 \\
     1553.1298  & -200.00   &   1.22 \\
     1582.0826  & -196.36   &   1.34 \\
     1583.0577  & -197.94   &   1.41 \\
     1585.0085  & -196.73   &   1.20 \\
     1585.0130  & -196.74   &   1.19 \\
     1586.0218  & -205.15   &   1.34 \\
     1679.8938  & -179.28   &   1.42 \\
     1702.9103  & -175.20   &   1.28 \\
     1703.8065  & -174.94   &   1.35 \\
     1704.8793  & -177.12  &    1.49 \\
     1705.8784  & -179.77  &    1.35 \\
     1706.8757  & -175.24  &    1.43 \\
     1755.7542  & -168.56  &    1.11 \\
     \hline
\end{tabular}}
\caption{}
\end{table}

\begin{table}[!ht]
\centerline{
\begin{tabular}{ccc}
\hline
\hline
HJD-2,450,000  &   RV [m~s$^{-1}$]    & Uncertainty [m~s$^{-1}$]   \\
\hline
\hline
     1884.1599  & -141.95  &    1.41 \\
     1898.1674  & -141.82  &    1.28 \\
     1899.1728  & -137.89  &    1.37 \\
     1900.1678  & -139.61  &    1.30 \\
     1901.1771  & -138.81  &    1.27 \\
     1972.1057  & -129.95  &    1.26 \\
     1973.1304  & -127.04  &    1.33 \\
     1982.1034  & -126.24  &    1.43 \\
     2030.9064  & -125.05  &    1.43 \\
     2101.8222  & -117.06  &    1.75 \\
     2334.1177   & -67.85   &   1.84 \\
     2712.0408   &  -0.36   &   1.37 \\
     3071.9391   &  54.34   &   1.56 \\
    \hline
    \multicolumn{3}{c}{RV Offset} \\
    \hline
     3480.0014   & 130.10   &   0.86 \\
     3480.0026   & 131.17   &   0.62 \\
     5232.1301   & 414.79   &  0.91 \\
     5973.1039   & 529.01   &   0.95 \\
     6109.7547   &  554.22  &   0.92 \\
     6319.0020   &  584.50  &  1.10 \\
     \hline
\end{tabular}}
\caption{Doppler radial velocity measurements for HD 114174. A horizontal line indicates the division between HIRES (detector) pre-upgrade and post-upgrade data points, requiring a differential offset for analysis. Uncertainties correspond to photon-noise alone.}
\end{table}

\subsection{High-Contrast Imaging}
We acquired first-epoch high-contrast images of HD~114174 on 2011 February 22  UT using NIRC2 and the Keck II AO system \citep{wizinowich_00}. The bright ($K_s=5.20$, \citealt{skrutskie_06}) star was placed behind the 300 mas diameter coronagraphic spot. We used the angular differential imaging (ADI) technique to discriminate between residual scattered starlight and companions through point-spread-function (PSF) subtraction \citep{marois_06}.

Raw frames were processed by flat-fielding the array, replacing bad pixels with interpolated values, and high-pass Fourier filtering. The latter step reduces the AO halo and thermal background (pedestal) while accentuating quasi-static speckles, which aids with image alignment. Frames were then multiplied by a gray-scale intensity mask (having contiguous transmission values between 0 and 1) to prevent the coronagraph from biasing image alignment. We used the Locally Optimized Combination of Images (LOCI) algorithm to increase effective contrast levels \citep{lafreniere_07}. The subtracted frames were then rotated to the standard East-North orientation using the parallactic angle to combine images and stack light from any off-axis sources.

Upon processing the full 2011 February data set, we noticed a faint candidate companion located $\approx 0.7\arcsec$ to the south in the final reduced image. Follow-up observations were taken on 2012 February 02, 2012 May 29, and 2012 July 4 UT with the $K'$ and $J$ filters to obtain color information and determine whether the candidate is associated with the primary. Fig. 2 shows the confirmation image taken one year after the first epoch.

We also obtained a deep ADI sequence of HD~114174 with NIRC2 in the L'-band on 2012 May 07 UT. These observations were recorded under good conditions ($\approx$0.5$\arcsec$ seeing), at low airmass (1.03-1.05), and achieved 40.5 degrees of field rotation, yet resulted in a non-detection, suggesting that HD~114174~B  may have blue colors. We derive a lower-limit to the brightness difference between HD~114174~B and its parent star of $\Delta L'>9.84$ mag. The companion was recovered at all other epochs using filters at shorter wavelengths.

\begin{figure}[!t]
\begin{center}
\includegraphics[height=2.8in]{f2.eps}
\caption{Confirmation image of HD~114174~B taken with NIRC2 at Keck. The $\Delta K'=10.73\pm0.12$ companion cannot be seen in individual raw frames with long exposures, but is clearly visible after PSF subtraction.}
\end{center}\label{fig:image}
\end{figure}

\section{PHOTOMETRY}
PSF subtraction modifies the measured flux (photometry) and location (astrometry) of faint companions that are initially buried in bright speckle noise \citep{crepp_11,pueyo_12}. We injected synthetic companions into the NIRC2 ADI data to estimate the amount of companion flux lost as a result of using the LOCI algorithm \citep{soummer_12}. Self-consistent introduction of fake companions into each frame also allows us to measure the intrinsic scatter in photometric and astrometric quantities and thus compute their uncertainty.

\begin{table*}[!ht]
\centerline{
\begin{tabular}{lc}
\hline
\multicolumn{2}{c}{HD~114174 Properties}     \\
\hline
right ascension [J2000]            &    13 08 51.02         \\
declination [J2000]                   &    +05 12 26.06        \\
$B$                                           &    7.47                     \\
$V$                                           &     6.80                     \\
$J_{\tiny{\rm 2MASS}}$            &    $5.613\pm0.026$  \\
$K_s$                                        &   $5.202\pm0.023$  \\
$J_{\tiny{MKO}}$                      &    $5.584\pm0.027$  \\
$K_{\tiny{MKO}}$                      &   $5.189\pm0.024$  \\
d [pc]                                          &    $26.14\pm0.37$   \\
proper motion [mas/yr]              &    $84.72\pm0.59$  E \\
                                                   &   $-670.11\pm0.47$ N  \\
\hline
\multicolumn{2}{c}{Host Star}      \\
\hline
Mass [$M_{\odot}$]           &    $1.05\pm0.05$        \\
Radius [$R_{\odot}$]         &     $1.06$       \\
Luminosity [$L_{\odot}$]    &   $1.13$         \\
Age [Gyr]                            &     $4.7^{+2.3}_{-2.6}$    \\
$\mbox{[Fe/H]}$                 &      $0.07\pm0.03$      \\
log g [cm $\mbox{s}^{-2}$]    &  $4.51\pm0.06$       \\
$T_{\rm eff}$ [$K$]             &    $5781\pm44$        \\
Spectral Type                     &       G5 IV-V               \\
v sini   [km/s]                        &   $1.8\pm0.5$        \\
\hline
\multicolumn{2}{c}{Companion$^a$}  \\
\hline
$\Delta J$                 &      $10.48\pm0.11$        \\
$\Delta K$                 &      $10.75\pm0.12$       \\
$\Delta L'$                 &        $>9.84$                            \\
$J$                              &     $16.06\pm0.11$         \\
$K$                             &    $15.94\pm0.12$          \\
$L$                             &    $>14.99$                              \\
$M_J$                     &    $13.97\pm0.11$          \\
$M_K$                    &     $13.85\pm0.12$        \\
$M_{L'}$                   &      $>12.90$                            \\
$T_{\rm eff}  [K]$         &      $8200\pm4000$      \\
$\log g$ [cm/s$^2$]        &          $8.90\pm0.02$        \\
$t_c$  [Gyr]                   &        $\approx3.4$           \\
$m_{\rm dyn}$ [$M_{\odot}$]        &    $>0.260\pm0.010$               \\
$m_{\rm model}$ [$M_{\odot}$]    &    $1.15\pm0.01$  \\
\hline
\end{tabular}}
\caption{(Top) Coordinates, apparent magnitudes, distance, and proper motion of HD~114174 from the SIMBAD database. Magnitudes are from 2MASS \citep{skrutskie_06}. Filter conversions from \citet{carpenter_01} have been applied to estimate stellar MKO values. The parallax-based distance is from {\it Hipparcos} measurements using the refined data reduction of \citealt{van_leeuwen_07}. (Middle) Host star physical properties are estimated from SME using HIRES template spectra as well as theoretical isochrones \citep{valenti_fischer_05}. (Bottom) Companion magnitude difference, apparent magnitude, absolute magnitude, and companion physical properties, such as the estimated cooling age ($t_c$), mass constraint (lower-limit) from dynamics ($m_{\rm dyn}$), and mass ($m_{\rm model}$) from photometry using white dwarf evolutionary models assuming a pure hydrogen atmosphere.}
\label{tab:starprops}
\end{table*}

We measured the relative brightness between HD~114174~B and its host star by comparing the flux from LOCI-processed images to unocculted images of the primary star, accounting for the difference in integration time. A fast-readout, subarray mode prevented HD~114174~A from saturating the array in unocculted frames. Subsequent reductions using fake companions and the same LOCI parameters were then performed to determine the (throughput) response of the data pipeline. The injected companions were placed at approximately the same angular separation as HD~114174~B, but with azimuthal separations sufficient to prevent artificial self-subtraction \citep{quanz_12}. Unocculted frames of the primary star informed our choice of the FWHM for fake companions.

Measurements using two different filters, $K'_{\tiny{MKO}}$ and $J_{MKO}$, provided relative flux and color information. Optimized for the local atmospheric transmission and thermal background on Mauna Kea \citep{tokunaga_02}, the $K'_{\tiny{MKO}}$ band is our filter choice for search-mode operation. We measure flux ratios of $\Delta K'_{\tiny{MKO}}=10.73\pm0.12$ and $\Delta J_{\tiny{MKO}}=10.48\pm0.11$ (Table 3). We then convert the $J_{\tiny{\rm 2MASS}}$ and $K_s$ magnitudes of the primary star to $J_{\tiny{MKO}}$ and $K_{\tiny{MKO}}$ using the $J_{\tiny{\rm 2MASS}}-K_s$ color from SIMBAD and empirical transformations from \citet{carpenter_01}.

To place the companion flux in the same filter basis\footnote{Central wavelength for the $K'_{\tiny{MKO}}$ and $K_{\tiny{MKO}}$ filters are $\lambda_{K'}=2.124\;\mu m$ and $\lambda_{K}=2.196\;\mu m$.}, we downloaded high-resolution spectra of various cold dwarfs from the IRTF spectral library\footnote{http://irtfweb.ifa.hawaii.edu/$\sim$spex/IRTF\_Spectral\_Library/} as well as the filter profiles from Keck Observatory\footnote{http://www2.keck.hawaii.edu/inst/nirc2old/filters.html}. IRTF library spectra from a G5 dwarf, HD 165185, served as a fiducial to compare the colors of the companion to the G5 primary, HD~114174~A. HD 165185 and HD~114174~A have $J-K_s$ colors consistent to within 0.045 mag \citep{skrutskie_06}, smaller than the 0.12 mag uncertainty in the measured companion-to-star flux difference.

Our initial interpretation of the companion's properties led us to believe it was a brown dwarf. We thus began filter conversion calculations using IRTF spectra from two cold brown dwarfs listed in \citet{cushing_05}, SDSS J125453.90-012247.4 (T2) and 2MASS J05591915-1404489 (T4.5). We first scaled the flux of the different T-dwarf spectra by a multiplicative factor to match our $\Delta K'_{\tiny{MKO}}$ measurement, then compared the $\Delta J_{\tiny{MKO}}$ model result to our $\Delta J_{\tiny{MKO}}$ measurement. The cooler T4.5 dwarf spectrum resulted in a significantly better fit to our measurements ($\Delta J_{\tiny{MKO}}$ within $1\sigma$) than the T2 dwarf spectrum which was discrepant at the $5\sigma$ level. Using the T4.5 spectrum reference, we find $\Delta K_{\tiny{MKO}}=10.75\pm0.12$ (from the measured $\Delta K'_{\tiny{MKO}}=10.73\pm0.12$), $K_{\tiny{MKO}}=15.94\pm0.12$, and $(J-K)_{\tiny{MKO}}=0.12\pm0.16$ (Table 3).

Both the absolute magnitude and colors of HD~114174~B are consistent with a cold brown dwarf (Fig. 3). However, as shown below (Section 5.1, Section 5.2), our dynamical analysis precludes this interpretation. Since the spectral energy distribution of cool white dwarfs and late-type brown dwarfs are both often dominated by collisionally induced $H_2$ absorption \citep{borysow_frommhold_90}, and the correction factor between the $K'_{\tiny{MKO}}$ and $K_{\tiny{MKO}}$ filters is small, particularly for dwarfs with neutral colors, we adopt the $\Delta K_{\tiny{MKO}}=10.75\pm0.12$ value for subsequent analysis.

\begin{figure}[!t]
\begin{center}
\includegraphics[height=4.3in]{f4.eps}
\caption{HD~114174~B absolute J-band magnitude and J-K color compared to cold brown dwarfs from \citet{leggett_10}. Black data points represent spectral types derived from moderate resolution spectroscopy (http://www.gemini.edu/staff/sleggett). Our $1\sigma$ measurement uncertainties from broadband photometry are shown as horizontal lines. HD~114174~B appears to be consistent with a T3.5 dwarf, but dynamical considerations indicate otherwise. Given the comparable near-infrared colors between brown dwarfs and white dwarfs, compact objects like HD~114174~B might be considered as a type of false-positive for high-contrast imaging surveys dedicated to searching for substellar companions.}
\end{center}\label{fig:image}
\end{figure}

\section{ASTROMETRY}
Our astrometric observations consist of four epochs taken over an 1.4 year baseline (Table~4). The proper-motion of HD~114174 is high, 675 mas $\mbox{yr}^{-1}$, allowing us to easily determine whether the companion shares the same space motion as the primary. We measured an accurate separation and position angle using the technique described in \citet{crepp_12a}. Following PSF subtraction, we fit Gaussian functions to the stellar and companion point-spread function to locate their centroids in each frame. The results from trials using numerous fake injected companions are averaged and uncertainty in the separation and position angle are initially taken as the measurement standard deviation. We then account for uncertainty in the plate scale and instrument orientation \citep{ghez_08} by propagating these errors to the final calculated position. Finally, a 5 mas uncertainty is added in quadrature to the rectilinear position of the star to account for unknown distortions introduced by the focal plane occulting mask (Q. Konopacky 2011, private communication). For companions with intensity comparable to, or fainter than, stellar quasi-static speckles using (near) Nyquist-sampled data, it is difficult to measure an astrometric position with precision much less than a single pixel \citep{marois_10_SOSIE,crepp_11}.

Fig.~4 shows multi-epoch astrometry measurements plotted against the expected motion of a distant background object. We find that HD~114174~B is clearly associated with HD~114174~A. The position of the companion has changed by only 28 mas over an 1.4 year time-frame. An unrelated background source placed at infinite distance with zero proper-motion would have moved relative to the host star by 902 mas over the same timeframe. HD~114174~B has a projected separation of $18.09\pm0.34$ AU (2012 July 4 UT) that appears to be decreasing with time.

\begin{table*}[!t]
\centerline{
\begin{tabular}{lccccc}
\hline
\hline
  Date [UT]      &   JD-2,450,000         &     Filter    &      $\rho$ [mas]      &    Position Angle [$^{\circ}$]     &   Projected Separation [AU]   \\
\hline
\hline
2011-02-22           &  5615.1           &             $K'$            &     $719.8\pm6.6$     &    $171.8^{\circ}\pm0.5^{\circ}$       &    $18.82\pm0.32$ \\
2012-02-02            &   5960.1            &            $K'$          &      $701.1\pm5.0$    &    $172.1^{\circ}\pm0.4^{\circ}$       &    $18.33\pm0.29$ \\
2012-05-29            &   6076.8            &            $K'$          &     $695.8\pm5.8$      &    $170.5^{\circ}\pm0.6^{\circ}$       &    $18.19\pm0.25$  \\
2012-07-04            &   6112.8            &            $ J$          &     $692.1\pm8.7$      &    $172.2^{\circ}\pm0.8^{\circ}$       &    $18.09\pm0.34$ \\
\hline
\hline
\end{tabular}}
\caption{Summary of astrometric measurements.}
\label{tab:astrometry}
\end{table*}

\begin{figure}[!t]
\begin{center}
\includegraphics[height=2.8in]{f3.eps}
\caption{Astrometric measurements demonstrating that HD 114174 B is co-moving with its host star. Location of the primary defines the coordinate system and is denoted by a star. The dashed curve shows the path that a distant background object would execute from 2011 February 22 UT through 2012 July 4 UT, accounting for proper-motion and parallactic motion.}
\end{center}\label{fig:astrometry}
\end{figure}

\section{COMPANION PROPERTIES}
\subsection{Mass Lower-limit from Dynamics}
A lower-limit to the companion mass may be determined from the RV acceleration and direct imaging angular separation \citep{torres_99,liu_02}. Although the HD~114174 Doppler signal already shows a hint of curvature (Fig. 1), we do not compute the full range of possible masses because the astrometry has yet to show significant orbital motion, which is {\it required} to place interesting constraints on orbital elements and the companion mass \citep{kalas_08,currie_11_betapic,crepp_12a,crepp_12b}. Instead, we evaluate the local slope of the RV time series (see also Section 5.4). We have obtained a recent Doppler RV observation (2013 January 26) to facilitate this calculation. Using an MCMC simulation to fit the instantaneous RV slope, we find a best-fitting Doppler acceleration of $57.4\pm0.4$ m$\:\rm{s^{-1} yr^{-1}}$, a value similar to that found in Section 2.1 when considering the full RV data set because curvature in the RV time-series is still subtle. This measurement translates to a minimum dynamical mass of $0.260\pm0.010M_{\odot}$, thus precluding the interpretation that HD~114174~B is substellar, despite its low luminosity and neutral colors.

\subsection{White Dwarf Interpretation}
White dwarfs are a common outcome of the evolutionary process for low-mass stars and are thus expected to comprise a non-negligible fraction of stellar companions \citep{holberg_98,breton_12}. With a sample of more than 100 targets, it is therefore not surprising that the TRENDS program would discover a compact object orbiting a nearby star. Indeed, it seems the only self-consistent interpretation of our photometric and dynamical measurements is that HD~114174~B is a white dwarf companion. Such serendipitous discoveries might be considered as direct imaging false-positives when searching for exoplanets and brown dwarfs \citep{zurlo_13}.

Despite the prevalence of white dwarfs throughout the galaxy \citep{oppenheimer_01}, relatively few benchmark compact objects have been discovered \citep{muirhead_13} and only several nearby companions with precise parallax measurements, such as Sirius B and Procyon B, have been imaged directly \citep{liebert_05}. Similar to field brown dwarfs, most detailed white dwarf studies involve age calibration based on stellar cluster membership \citep{williams_04}, and those with reliable ages rarely have simultaneous dynamical mass estimates \citep{girard_00,bond_09}. HD~114174 thus represents a convenient system for studying the intricacies of post-main-sequence stellar evolution at low mass.

Without spectra, we cannot determine the chemical composition of HD~114174~B. Instead, we proceed by considering two different canonical cases: pure hydrogen and pure helium white dwarf atmospheres. By fitting theoretical models to our $J$ and $K$ observations, we can constrain the effective temperature, mass, and cooling age of HD~114174~B under reasonable assumptions about its composition.

\subsection{Synthetic Photometry \& Fitting}
For pure hydrogen atmospheres and synthetic spectra, we use the theoretical models of \citet{tremblay_11}.\footnote{http://www.astro.umontreal.ca/$\sim$bergeron/CoolingModels/} The models include improved calculations for Stark broadening of hydrogen lines \citep{tremblay_09} and non-ideal perturbations, described within the occupation probability formalism of \citet{hummer_88}, directly inside the line profile calculations. The models also include Lyman alpha profile calculations from \citet{kowalski_06}. Convective energy transport is taken into account following the revised ML2/$\alpha$ = 0.8 prescription of mixing-length theory from \citet{tremblay_10}. Details regarding our helium atmosphere models are provided in \citet{bergeron_11}. Model grids cover a range in effective temperature between $T_{\rm eff}~= 1500$ K and 40,000 K in steps varying from 500~K to 5000~K. The surface gravity ranges from $7.0 \leq \log g \leq 9.0$ by steps of 0.5 dex. Synthetic colors are obtained using the procedure of \citet{holberg_06} based on Vega fluxes taken from \citet{bohlin_04}.

\begin{figure}[!t]
\begin{center}
\includegraphics[scale=0.52,bb= 150 210 420 560]{f5.eps}
\caption{White dwarf model fits to our JKL' photometry assuming pure hydrogen (top) and pure helium (bottom) atmospheres. The observed JK fluxes are represented by errorbars. Model monochromatic fluxes are shown as the solid red line. Our measurements yield physical parameter estimates as listed in Table 3. The L'-band non-detection from Keck is shown as an upper-limit.}
\end{center}\label{fig:wdmodel}
\end{figure}

The method used to fit photometric data is similar to that described in \citet{bergeron_01}, which we briefly summarize. We first transform the magnitudes in each bandpass into observed average fluxes $f_{\lambda,m}$ using the following equation:
\begin{equation}
  m = -2.5 \log f_{\lambda,m} + c_{m},
\label{eq:1}
\end{equation}
\noindent where
\begin{equation}
  f_{\lambda,m} = \frac{\int^{\infty}_{0}f_{\lambda}S_{m}(\lambda)d\lambda}{\int^{\infty}_{0}S_{m}(\lambda)d\lambda}.
\label{eq:2}
\end{equation}

\noindent The transmission functions $S_{m}(\lambda)$ along with the constants $c_{m}$ for each bandpass are described at length in \citet{holberg_06} and references therein. To make use of all photometric measurements simultaneously, we convert measured magnitudes into observed fluxes using equation (\ref{eq:1}) and compare the resulting energy distributions with those predicted from our model atmosphere calculations. Model fluxes are also averaged over filter bandpasses by substituting $f_{\lambda}$ in equation (\ref{eq:2}) for the monochromatic Eddington flux $H_{\lambda}$. The average observed fluxes $f_{\lambda,m}$ and model fluxes $H_{\lambda,m}$, which depend on $T_{\rm eff}$, $\log g$ and $N$(He)/$N$(H), are related by
\begin{equation}
  f_{\lambda,m} = 4\pi(R/d)^{2}H_{\lambda,m}
\label{eq:3}
\end{equation}
\noindent where $R/d$ is the ratio of the radius of the star to its distance from Earth.

Our fitting procedure relies on the nonlinear, least-squares, steepest descent method of Levenberg-Marquardt. The value of $\chi^{2}$ is taken as the sum over all bandpasses of the difference between both sides of equation (\ref{eq:3}), properly weighted by the corresponding observational uncertainties. In our fitting procedure, we consider only $T_{\rm eff}$~and the solid angle, $R/d$, as free parameters. We begin by guessing a value for the surface gravity ($\log g\;=\;8$) and determine the corresponding effective temperature and solid angle. Combined with the distance,  $d$, obtained from the trigonometric parallax measurement, this yields directly the radius of the star $R$. The radius is then converted into mass using evolutionary models similar to those described in \citet{fontaine_01} but with C/O cores, $q$(He) $\equiv$ $\log M_{He}/M_{\star} = 10^{-2}$ and $q(H) = 10^{-4}$, which are representative of hydrogen atmosphere white dwarfs, and $q(He) = 10^{-2}$ and $q(H) = 10^{-10}$, which are representative of helium atmosphere white dwarfs. In general, the $\log g$ value obtained from the inferred mass and radius ($g = GM/R^{2}$) will be different from our initial guess of $\log g=8$, and the fitting procedure is thus repeated until an internal consistency in $\log g$ is reached.

Assuming a hydrogen-dominated atmosphere, we find: $T_{\rm eff} =  8200\pm4000$ K and $\log g=8.90\pm0.02$, which corresponds to a cooling age of $t_c\approx3.4$ Gyr and model-dependent mass $m_{\rm model}=1.15\pm0.01M_{\odot}$. Assuming a Helium-dominated atmosphere, we find: $T_{\rm eff}=7700\pm4200$ K, $\log g=8.78\pm0.02$, $t_c\approx3.5$ Gyr, and $m_{\rm model}=1.08\pm0.01M_{\odot}$. With negligible differences between the physical parameter determinations when considering both models, we adopt the pure-H model values (Table 3). An $m=1.1M_{\odot}$ remnant corresponds to an $\approx$$6.2M_{\odot}-6.9M_{\odot}$ progenitor mass \citep{kalirai_08,williams_09}. Thus, HD~114174~B spent only a short period of time ($\approx 50$ Myr) on the main-sequence.

\subsection{Period Range from Dynamics and Estimated Companion Mass}
With an estimate of the mass of HD~114174~B, we can constrain the system orbital period using dynamics. We first determine the true (physical) separation of HD~114174~B from HD~114174~A using the instantaneous Doppler acceleration ($57.4\pm0.4$ m$\:\rm{s^{-1} yr^{-1}}$) and measured angular separation ($692.1\pm8.7$ mas) from geometry (e.g., \citealt{howard_10}). We find that HD~114174~B is presently $59.8\pm0.4$ AU from its host star, much further than the projected separation ($18.09\pm0.34$ AU). The true anomaly cannot be determined from the current data set. However, assuming the orbit configuration is near apastron or periastron, we can constrain the semimajor axis and thus orbital period. We find that the period is between 154--881 years for eccentricities in the range $0 \leq e \leq0.5$. The period lower-limit corresponds to an expected RV semi-amplitude of $K\approx4400$ m/s assuming an edge-on orbit, consistent with Fig. 1.

Our numerical simulations indicate that $\approx5-10$\% of an orbit cycle is nominally required following a direct imaging discovery to place interesting constraints on all six orbit elements and the companion mass using dynamics for astrometric and Doppler RV signal-to-noise levels comparable to those of HD~114174~B \citep{crepp_2012AAS}. Thus, we may be able to determine the system orbit and mass of HD~114174~B as early as $\approx$8 years from now with sufficient observing cadence.

\section{SUMMARY \& DISCUSSION}
We report the discovery of a faint companion orbiting the nearby G5 star HD~114174. Initially identified as a promising high-contrast imaging target due to a measured long-term RV acceleration, we have used NIRC2 at Keck to obtain direct images of the source that is responsible for causing the Doppler trend. Multi-epoch detections in two different filters demonstrate association through common proper-motion over an 1.4 year time frame. The RV data spans 16.0 years and shows curvature (a change in the acceleration). Since HD~114174 has a precise parallax measurement, it will be possible to place strong constraints on the companion orbit (all six parameters) and mass as soon as the direct imaging astrometry shows curvature.

Our broadband photometric measurements indicate that the companion has neutral, or possibly blue colors ($J-K=0.12\pm0.16$) suggesting interesting atmospheric physics. Deep ADI observations in the {\it L'}-band under good seeing conditions at Keck resulted in a non-detection, further supporting the interpretation of a decreasing NIR spectral-energy distribution with increasing wavelength. HD~114174~B has an absolute magnitude ($M_J=13.97\pm0.11$) and colors consistent with a T3-dwarf. However, dynamical considerations preclude the interpretation of a substellar companion. We calculate a firm minimum mass of $m>0.260\pm0.010M_{\odot}$ using presently available RV and imaging data.

HD~114174~B appears to be a white dwarf. Using theoretical models to fit our photometry and assuming a pure hydrogen atmosphere, we estimate $T_{\rm eff} =  8200\pm4000$ K and $\log g=8.90\pm0.02$, which corresponds to a cooling age of $t_c\approx3.4$ Gyr and remnant mass $m_{\rm model}=1.15\pm0.01M_{\odot}$, consistent with the $4.7^{+2.3}_{-2.6}$ Gyr age estimate of the system based on isochronal analysis of HD~114174~A, and also consistent with the dynamical mass lower limit. Calculations assuming a Helium-rich atmosphere yield similar results.

Uncertainty in $T_{\rm eff}$ is presently limited by the precision of our $J$ and $K$ photometry and current wavelength coverage. Further, the NIR colors of cool white dwarfs are known to exhibit an inversion (from having red colors to blue colors as $T_{\rm eff}$ decreases) as the result of collisionally-induced H$_2$ absorption. Thus, there exists a degeneracy in $T_{\rm eff}$ for $J-K\approx0.0$, possibly complicating the interpretation of HD~114174~B's spectrophotometric signal at the lowest temperatures currently allowed by our models. For example, the white dwarf companion Gliese 86 B ($J-K=1.0\pm0.3$) was initially characterized as a brown dwarf \citep{els_01,mugrauer_05}. Deeper L'-band observations obtained with the LBT \citep{skrutskie_10,skemer_12} and direct NIR spectroscopy measurements using an integral-field unit \citep{hinkley_11_PASP} will allow us to refine HD~114174~B's physical properties. Furthermore, it may be possible to study HD~114174~B at shorter wavelengths using HST in a fashion similar to that of \citet{barstow_05} who obtained a spectrum of Sirius B with STIS.

HD~114174 A is a marginally evolved Sun-like star located at $d=26.14\pm0.37$ pc. Thus, it may be possible to measure its radius directly using CHARA interferometry, placing tighter constraints on the primary star's mass and hence the system age and also secondary mass \citep{boyajian_09}. A similar analysis has been performed for the benchmark brown dwarf HR~7672 B \citep{crepp_12a}. In the case of a white dwarf, a more accurate mass and age will likewise constrain physical properties of the progenitor. The HD~114174 system thus represents a useful testbed for studying white dwarf initial-to-final mass ratios at a level of detail comparable to Sirius and Procyon. Given the anticipated sensitivity of forth-coming high-contrast imaging programs that use ``extreme" AO, it is likely that additional benchmark white dwarf companions will be (accidentally) uncovered in the near future. A more complete census of Sirius-like systems may ultimately be undertaken with the GAIA space mission using precision astrometry.

\section{ACKNOWLEDGEMENTS}
We thank Sasha Hinkley for trading NIRC2 observing time that allowed us to acquire observations of HD~114174~B in a complementary bandpass, demonstrating its unusual colors, and Mike Liu for pointing out reference to the Gliese 86 system which shares a similar story as HD~114174~B for being a white dwarf initially characterized as a brown dwarf. The referee, John Subasavage, provided helpful comments that improved the clarity of our manuscript. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. The TRENDS high-contrast imaging program is supported by NASA Origins grant NNX13AB03G. JAJ is supported by generous grants from the David and Lucile Packard Foundation and the Alfred P. Sloan Foundation.

Data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science, and the Pennsylvania Space Grant Consortium.

\begin{small}
\bibliographystyle{jtb}


\section{Introduction}

This paper provides two homologically motivated generalizations of a module-theoretic result proved by \textsc{Napp Avelli}, \textsc{van der Put}, and \textsc{Rocha}. This result was expressed in \cite{NR10} using the dual system-theoretic language and applied to behavioral control.
Their algebraic proof covers at least the polynomial ring $R=k[x_1,\ldots,x_n]$ and the \textsc{Laurent} polynomial ring $R=k[x_1^\pm,\ldots,x_n^\pm]$ over a field $k$.
The corresponding module-theoretic statement is the following:

\begin{theorem} \label{coro:diego}
  Let $R$ be one of the above rings and $M = R^q / A$ a finitely generated \emph{torsion-free} module.
  If there exists a submodule $B \leq R^q$ such that $A \cap B = 0$ and $T := R^q/(A+B)$ is of codimension at least $2$ then $M$ is free.
\end{theorem}

In fact, they prove a more general statement of which the previous is obviously a special case.
However, the special statement implies the more general one.

\begin{theorem}[{\cite[Theorem~18]{NR10}}] \label{thm:diego}
  Let $R$ be one of the above rings and $M = R^q / A$ a finitely generated module.
  If there exists a submodule $B \leq R^q$ such that $A \cap B = 0$ and $T := R^q/(A+B)$ is of codimension at least $2$ then the torsion-free factor $M/ \tor(M)$ of $M$ is free.
\end{theorem}

In the proposed module-theoretic generalization of Theorem~\ref{coro:diego} the notions ``torsion-free'', ``codimension'' and ``free'' are replaced by the more homological notions ``torsionless'', ``grade'', and ``projective'', respectively.

We start by describing the very basics of the duality between linear systems and modules in Section~\ref{sec:duality}.
The two notions ``torsionless'' and ``grade'' are briefly recalled in Section~\ref{sec:torsionless_grade}.
In Section~\ref{sec:module-theoretic_generalization1} a module-theoretic generalization of Theorem~\ref{coro:diego} is stated and proved.
The proof relies on an \textsc{Abel}ian generalization which is treated in Section~\ref{sec:Abelian_generalization}.
Since torsion-freeness admits a system-theoretic interpretation we need to discuss the relation between being torsion-free and being torsionless to justify the word ``generalization''.
Indeed, torsionless modules are torsion-free but the converse is generally false (cf.~Remark~\ref{rmrk:Q_not_torsionless} for a precise statement).
Section~\ref{sec:torsionless=torsion-free} describes a fairly general setup in which the converse does hold.
And only when it holds are we able to prove the corresponding generalization of Theorem~\ref{thm:diego}.
This is done in Section~\ref{sec:module-theoretic_generalization2}.
Finally, Appendix~\ref{sec:converse_key_lemma} contains a converse to the key Lemma of this paper.

\medskip
\textbf{Convention:} Unless stated otherwise $R$ will always denote a not necessarily commutative unitial ring.
The term ``domain'' will not imply commutativity.

Everything below is valid for left and for right $R$-modules.

\section{Duality between linear system theory and module theory} \label{sec:duality}

For an $R$-module $\F$ we define the category of \textbf{$\F$-behaviors} as the image of the contravariant $\Hom$-functor $\Hom_R(-,\F): R\mathrm{-Mod} \to \mathrm{Mod-}C$, where $C$ is the center of $R$ (or the endomorphism ring of $\F$ or any unitial subring of thereof).
$R$ is called the ring of functional operators and $\F$ a signal module or signal space.

An $R$-module $M$ is said to be cogenerated by $\F$ if $M$ can be embedded into a direct power $\F^I$ for some\footnote{One can take the index set to be the solution space $I := \Hom_R(M,\F)$ and require in the definition that the evaluation map from $M$ to the direct power $\F^I$, sending $m \in M$ to the map $I \to \F, \phi \mapsto \phi(m)$, is an embedding.} index set $I$.
$\F$ is a called a cogenerator if it cogenerates any $R$-module $M$, or, equivalently, if the duality functor $\Hom_R(-,\F)$ is faithful.
In particular, a cogenerator is a faithful $R$-module.
The duality functor $\Hom_R(-,\F)$ is exact if and only if $\F$ is injective.
An injective $\F$ is a cogenerator if and only if the solution space $\Hom_R(M,\F) \neq 0$ for each\footnote{Cyclic or even simple $R$-modules suffice (cf.~\cite[Theorem~3.1]{TI64}).} $M \neq 0$.
In particular, all simples can be embedded into an injective cogenerator.
Summing up, $\Hom_R(-,\F)$ is exact and reflects exactness\footnote{This follows easily from the fact that an exact faithful functor of \textsc{Abel}ian categories is conservative, i.e., reflects isomorphisms (see, e.g., \cite[Lemma~A.1]{BL_SerreQuotients}).
An exact functor of \textsc{Abel}ian categories which also reflects exactness is called ``faithfully exact'' in  \cite[Definition~1]{TI64}.
Injective cogenerators are called ``faithfully injective'' in \cite[Definition~3]{TI64}.} (and hence faithful) if and only if $\F$ is an injective cogenerator.
In this case the $\Hom$-duality between $R$-modules and $\F$-behaviors is perfect.
The above statements are true in any \textsc{Abel}ian category with products \cite[§~IV.6]{Sten75}.

The \textsc{Abel}ian group $\mathbb{Q}/\mathbb{Z}$ of characters of $\mathbb{Z}$ is an injective cogenerator in the category of \textsc{Abel}ian groups.
Likewise, the $R$-module $\Hom_\mathbb{Z}(R,\mathbb{Q}/\mathbb{Z})$ is called the module of characters of $R$ and is an injective cogenerator in $R\mathrm{-Mod}$.
This follows from the adjunction between $\Hom$ and the tensor product functor \cite[Proposition~I.9.3]{Sten75}.
The $k$-dual $R^\vee := \Hom_k(R,k)$ is an injective cogenerator for each $k$-algebra $R$ over a field $k$.
This classical result was already used in \cite[Corollary 3.12, Remark 3.13]{OB}.
\textsc{Plesken} and \textsc{Robertz} gave a constructive proof for the injectivity of the $k$-dual $R^\vee$ when $R$ is a multiple \textsc{Ore} extension over a computable field $k$ admitting a \textsc{Janet} basis notion  (cf.~\cite[Corollary 4.3.7, Theorem 4.4.7]{robphd}).
Furthermore, a minimal injective cogenerator always exists \cite[Proposition~19.13]{Lam} (see \cite[Subsection~19A]{Lam} for more details on injective cogenerators).

However, only those injective cogenerators which can be interpreted as a space of ``generalized functions'' (like distributions, hyperfunctions, microfunctions) are of direct significance for system theory in the engineering sense.
\textsc{Oberst} considers in \cite{OB} injective cogenerators $\F$ over commutative \textsc{Noether}ian rings which are large, i.e., satisfying $\Ass(\F)=\Spec(R)$.
\textsc{Fröhler} and \textsc{Oberst} prove in \cite{FO98} that the space of \textsc{Sato} hyperfunctions on an open interval $\Omega \subset \mathbb{R}$ is an injective cogenerator for the noncommutative ring $R:=A\left[\frac{\mathrm{d}}{\mathrm{d}t}\right]$ where $A := \left\{\frac{f}{g} \mid f,g \in \mathbb{C}[t], \forall \lambda \in \Omega: g(\lambda) \neq 0 \right\}$.
\textsc{Zerz} shows in \cite{Zer06} that the space of $\mathbb{R}$-valued functions on $\mathbb{R}$ which are smooth except at finitely many points\footnote{As a referee remarked, this restriction rules out singularities which solutions of ODEs with varying coefficients might generally exhibit.} is an injective cogenerator for the rational \textsc{Weyl} algebra $B_1(\mathbb{R})=\mathbb{R}(t)[\frac{\mathrm{d}}{\mathrm{d}t}]$.

From now on let $\F$ be an injective cogenerator with system-theoretic relevance.
Restricting to factor modules $M=R^q/A$ of a fixed free module $R^q$ yields a (non-intrinsic) \textsc{Galois} duality between the submodules $A$ of $R^q$, the so-called equations submodules, and $\F$-behaviors $\mathscr{M} = \Hom_R(R^q/A, \F)$.

In system-theoretic terms a factor module of the module $M$ corresponds to a subbehavior of $\mathscr{M} = \Hom_R(M,\F)$, and the torsion-free factor to the largest controllable subbehavior.
All degrees of torsion-freeness (including reflexivity and projectivity) are related to successive parametrizability of multi-dimensional systems in \cite{PQ99,CQR05}.
Freeness of modules corresponds to flatness of linear systems \cite{Fl}.
A common factor module $T$ of $M=R^q / A$ and $N=R^q / B$ corresponds to the so-called interconnection, i.e., the intersection $\mathscr{M} \cap \mathscr{N}$ of the two behaviors $\mathscr{M},\mathscr{N}$ corresponding to $M,N$.
The interconnection is called regular when $A \cap B = 0$.
Finally, the codimension of a module corresponds to the degree of autonomy of the corresponding behavior.
This paper suggests, in particular, the use of grade as a substitute for codimension to define the degree of autonomy in the noncommutative setting.

\section{Torsionless modules and grade} \label{sec:torsionless_grade}

We will use the notion of a torsionless module, due to \textsc{H.~Bass}, to provide a natural module-theoretic generalization of Theorem~\ref{coro:diego}.

\begin{defn}
  An $R$-module $M$ is called \textbf{torsionless} if it is cogenerated by the free module $R$, i.e., if it can be embedded into a direct power $R^I := \prod_{i \in I} R$, for some index set $I$.
\end{defn}

\begin{rmrk} \label{rmrk:torsionless}
From the definition we conclude that:
  \begin{enumerate}
    \item Any submodule of a torsionless module is torsionless and any direct product (and hence sum) of torsionless modules is torsionless.
    \item Since direct sums embed in direct products any submodule of a free module is torsionless.
      Thus, projective modules and left and right ideals are torsionless. \label{rmrk:torsionless.projective}
  \end{enumerate}
\end{rmrk}

We denote by $M^*:=\Hom_R(M,R)$ the $R$-dual of an $R$-module $M$.
It is easy to see that $M$ is torsionless iff\footnote{
The ``only if''-part follows by setting $\lambda$ to be the composition of the embedding $\jmath:M \hookrightarrow R^I$ and the projection $\pi_\iota: R^I \to R$ such that $\pi_\iota(\jmath(m)) \neq 0$.
The ``if''-part follows by setting $I=M^*$ and $\jmath$ to be the evaluation map $\epsilon_M: M \to M^{**}, m \mapsto (\lambda \mapsto \lambda(m))$  considered as a map to $R^I \supset M^{**}$.} for any $m \in M \setminus \{0\}$ there exists a functional $\lambda \in M^*$ such that $\lambda(m) \neq 0$.
Hence, $M$ is torsionless iff the natural evaluation map
\[
  \epsilon_M: M \to M^{**}, m \mapsto (\lambda \mapsto \lambda(m))
\]
is a monomorphism\footnote{Recall, $M$ is called  \textbf{reflexive} if $\epsilon_M$ is an isomorphism.}.
The dualized evaluation map $\epsilon_M^*:M^{***} \to M^*$ is a post-inverse of the evaluation map of the dual module $\epsilon_{M^*}:M^* \to M^{***}$, i.e., the latter is a split monomorphism (cf.~\cite[Remark~(4.65).(f)]{Lam99}).
In particular, the dual $M^*$ and the double-dual $M^{**}=(M^*)^*$ are torsionless modules.
This gives rise to the following definition.

\begin{defn}
  The \textbf{torsionless factor} of an $R$-module $M$ is the coimage $M/\ker \epsilon_M$ of the evaluation map.
\end{defn}

\begin{rmrk} \label{rmrk:Q_not_torsionless}
  If $R$ is a domain then any torsionless module is torsion-free.
  The converse is false: The infinitely generated $\mathbb{Z}$-module $\mathbb{Q}$ is torsion-free with a zero evaluation map, i.e., the ``opposite'' of being torsionless.
  Finitely generated modules behave better in this respect (cf.~Theorem~\ref{thm:torsionless=torsion-free}).
  While submodules of torsion modules are torsion, the torsionless $\mathbb{Z}$-submodule $\mathbb{Z} \leq \mathbb{Q}$ shows that having a zero evaluation map is not stable under passing to submodules.
  Still, the factor module $\mathbb{Q}/\mathbb{Z}$ has a zero evaluation map.
\end{rmrk}

Recall, an $R$-module $T$ is is said to have \textbf{grade} at least\footnote{This is a more convenient than defining the grade by an equality.} $c$ if $\Ext^i(T,R)=0$ for all $i<c$.
The grade of the associated cyclic module $R/\Ann(T)$ coincides with the grade of $T$.

\begin{rmrk} \label{rmrk:CM}
  Let $R$ be a commutative \textsc{Noether}ian ring.
  The grade of an $R$-module $T$ coincides, by a theorem of \textsc{Rees}, with $\depth \Ann(T) := \depth(\Ann(T),R)$ \cite[Proposition~18.4]{eis}, \cite[Theorem~1.2.5]{BrunsHerzog}.
  $R$ is called \textsc{Cohen-Macaulay} if the notions of codimension and grade coincide, i.e., if $\codim T := \codim \Ann(T)$ coincides with $\grade T = \depth \Ann(T)$ for all modules $T$ \cite[Introductions to Chapters~9 and~18]{eis}.
  The reader is referred to \cite[Part~II]{BrunsHerzog} for large classes of \textsc{Cohen-Macaulay} rings.
\end{rmrk}

The following definition is used to formulate the module-theoretic generalization of Theorem~\ref{coro:diego}.

\begin{defn} \label{defn:projective_up_to_grade}
  We say that an $R$-module $M$ is \textbf{projective up to grade $c$} if there exists a projective module $P$ and an epimorphism $P \stackrel{\pi}{\twoheadrightarrow} M$ such that $A := \ker \pi \leq P$ admits \textbf{a complement up to grade $c$ in $P$}, i.e., if there exists a submodule $B \leq P$ with $A \cap B=0$ and $T:=P / (A + B)$ has grade at least $c$.
  If $M$ is finitely generated then we insist that $P$ is finitely generated.
\end{defn}

\section{A module-theoretic generalization of Theorem~\ref{coro:diego}} \label{sec:module-theoretic_generalization1}

Projective modules are torsionless (Remark~\ref{rmrk:torsionless}.\eqref{rmrk:torsionless.projective}) and obviously projective up to grade $c$, for any $c$.
The converse is true for finitely generated modules and $c \geq 2$, yielding a module-theoretic generalization of Theorem~\ref{coro:diego}:
\begin{theorem} \label{coro:main1}
  Let $R$ be a ring and $M$ a finitely generated $R$-module.
  If $M$ is torsionless and projective up to grade $2$ then $M$ is projective.
\end{theorem}
\begin{proof}
  Let $P \stackrel{\pi}{\twoheadrightarrow} M$ be the f.g.~projective module of Definition~\ref{defn:projective_up_to_grade}, $A := \ker \pi \leq P$, $B$ be a complement up to the grade $2$ of $A$ in $P$, and $T := P / (A + B)$ the factor module of grade $\geq 2$, i.e., $\Hom(T,R)=0=\Ext^1(T,R)$.
  The assertion will follow from Theorem~\ref{thm:main1} as soon as we have shown that $\Hom(T,M) = 0 = \Ext^1(T,P)$ which we will do now: \\
  Since $M$ is torsionless there exists an embedding\footnote{Proposition~\ref{prop:embed_in_free} provides an alternative embedding into a free module of finite rank.} $\jmath: M \hookrightarrow R^I$ in a direct product for some index set $I$.
  As the left exact covariant $\Hom$-functor commutes with direct products \cite[Proposition~I.3.5]{HS} it follows that
  \[
    \Hom(T,M) \cong \Hom(T, \jmath(M)) \leq \Hom(T,R^I) \cong \Hom(T,R)^I=0 \mbox{.}
  \]
  And since $P$ is finitely generated projective it is a direct summand of a free module $R^p \cong P \oplus P'$ of finite rank $p$.
  Finally, the additivity of $\Ext^1(T,-)$ yields
  \[
    \Ext^1(T,P) \leq \Ext^1(T,P) \oplus \Ext^1(T,P') \cong \Ext^1(T,R^p) = \Ext^1(T,R)^p = 0 \mbox{.} \qedhere
  \]
\end{proof}

\section{An \textsc{Abel}ian generalization of Theorem~\ref{coro:diego}} \label{sec:Abelian_generalization}

\begin{wrapfigure}[6]{r}{2.7cm}
\vskip -0.8cm
\begin{tikzpicture}

  \coordinate (dl) at (-1,-1);
  \coordinate (dr) at (1,-1);

  \node (T) {$T$};
  \node (M) at (dl) {$M$};
  \node (N) at (dr) {$N$};
  \node (P) at ($(M)+(dr)$) {$P$};
  \node (A) at ($(P)+(dr)$) {$A$};
  \node (B) at ($(P)+(dl)$) {$B$};

  \draw [->>] (M) -- (T);
  \draw [->>] (N) -- (T);
  \draw [->>] (P) -- (M);
  \draw [->>] (P) -- (N);
  \draw [left hook->] (A) -- (P);
  \draw [right hook->] (B) -- (P);

\end{tikzpicture}
\end{wrapfigure}
Let $\A$ be an \textsc{Abelian} category and $P \cong M \times_T N \in \A$ a subdirect product\footnote{Also called fiber product.}  of two objects $M$ and $N$ over a common factor object $T$, i.e., $M \twoheadleftarrow P \twoheadrightarrow N$ is the pullback of the two epis $M \twoheadrightarrow T \twoheadleftarrow N$.
\begin{theorem} \label{thm:main1}
  If $\Hom(T,M) = 0 = \Ext^1(T,P)$ then the epi $P \twoheadrightarrow M$ is split and $M$ is isomorphic to a direct summand of $P$.
  If furthermore $P$ is projective then so is $M$.
\end{theorem}

The following simple lemma is the essence of the short proof of Theorem~\ref{thm:main1}.
We keep the above notation and set $A := \ker \left( P \twoheadrightarrow M \right)$ and $B := \ker\left( P \twoheadrightarrow N \right)$.

\begin{lemm} \label{lemm:key}
  If $\Ext^1(T,A) = 0$ then $A$ has a complement $B'\cong M$ in $P$ which contains $B$.
  In particular, $M$ is isomorphic to a direct summand of $P$.
\end{lemm}

\vskip -0.4cm
\begin{wrapfigure}[8]{r}{4.7cm}
\vskip -0.55cm
\begin{tikzpicture}[C/.style={color=red,line width=1.5pt,dotted},new/.style={color=blue,line width=2pt},map/.style={color=gray,dashed}]

  \coordinate (d) at (0,-1);
  \coordinate (dl) at (-1.5,-1.5);
  \coordinate (l) at (-1,0);
  \coordinate (dr) at (1,-1);

  \coordinate (P);
  \coordinate (B') at (dr);
  \coordinate (S) at (d);
  \coordinate (A) at ($(S)+(dl)$);
  \coordinate (B) at ($(S)+(dr)$);
  \coordinate (0) at ($(A)+(dr)$);

  \fill (P) circle (2pt);
  \fill (B') circle (2pt);
  \fill (S) circle (2pt);
  \fill (A) circle (2pt);
  \fill (B) circle (2pt);
  \fill (0) circle (2pt);

  \draw (0) -- (A) -- (S) -- (P);
  \draw (0) -- (B) -- (S);
  \draw (B) -- (B') -- (P);
  \draw[decorate,decoration=brace] ($(S)+0.1*(l)-0.1*(d)$) -- node [left] {\small $T$} ($(P)+0.1*(l)+0.1*(d)$);
  \draw[decorate,decoration=brace] ($(A)+0.2*(l)$) -- node [left] {\small $M$} ($(P)+1.7*(l)$);
  \draw[decorate,decoration=brace] ($(P)-1.7*(l)$) -- node [right] {\small $N$} ($(B)-0.7*(l)$);
  \draw[dotted,gray,thick] ($(P)+1.5*(l)$) -- ($(P)+0.22*(l)$);

  \draw[dotted,gray,thick] ($(P)-1.5*(l)$) -- ($(P)-0.22*(l)$);
  \draw[dotted,gray,thick] ($(B)-0.5*(l)$) -- ($(B)-0.22*(l)$);

  \node at (P) [above right=-0.1em] {\small $P$};
  \node at (B') [right] {\small $B'$};
  \node at (S) [left=0.4em] {\small $S$};
  \node at (A) [below left] {\small $A$};
  \node at (B) [below right] {\small $B$};

\end{tikzpicture}
\end{wrapfigure}
\mbox{}
\begin{proof}
  Set $S:=A+B \leq P$, the direct sum of $A$ and $B$.
  The assumption $\Ext^1(T,A) = 0$ and the natural isomorphism $S/B \cong A$ imply that the short exact sequence $0 \to S/B \to N \to T \to 0$ splits.
  In other words, there exists a subobject $B'$ of $P$ with $B' \geq B$ such that $B'/B$ is a complement of $S/B \cong A$ in $P/B$.
  Since $B' \cap (A+B) = B$ it follow that $B'$ is a complement of $A$ in $P$, canonically isomorphic to $M$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main1}]
  To apply Lemma~\ref{lemm:key} we need to show that $\Ext^1(T,A) = 0$.
  Indeed, $\underbrace{\Hom(T,M)}_{=0} \to \Ext^1(T,A) \to \underbrace{\Ext^1(T,P)}_{=0}$ is part of the long exact $\Ext(T,-)$-sequence with respect to the short exact sequence $0 \to A \to P \to M \to 0$.
  Hence, $\Ext^1(T,A) = 0$.
\end{proof}

Lemma~\ref{lemm:key} has an interesting converse which we did not need here.
It is treated in Appendix~\ref{sec:converse_key_lemma}.

\section{When does torsion-free imply torsionless?} \label{sec:torsionless=torsion-free}

In this section we assume $R$ to be two-sided \textsc{Noether}ian\footnote{$R$ two-sided coherent is, as usual, enough but we stick to two-sided \textsc{Noether}ian for lack of references.}.
A \textbf{finite projective presentation} of a f.g.~$R$-module $M$ is an exact sequence
 $M \twoheadleftarrow P_0 \xleftarrow{\partial} P_1$ with f.g.~projective $R$-modules $P_0$ and $P_1$.
For such a module define the \textbf{\textsc{Auslander} dual} $A(M)$ to be the cokernel of the dual (or pullback) map $\partial^*:P_0^* \to P_1^*$.
Like the syzygy modules of $M$, the \textsc{Auslander} dual is well-defined up to projective equivalence.
In particular $\Ext^i(A(M),R)$ does not depend on the finite projective presentation for $i>0$.
Furthermore, if $M$ is projective then $A(M)=0$ (up to projective equivalence) and $\Ext^i(A(M),R)=0$ for all $i>0$ (for a converse statement cf.~\cite[Theorem~7]{CQR05}).

The kernel and cokernel of the evaluation map $\epsilon:M \to M^{**}$ were characterized by \textsc{Auslander}, where $M$ is assumed to have a finite projective presentation.
As one of many applications of his theory of coherent functors \cite{A} he proved the existence of a natural monomorphism $\tau: \Ext^1(A(M),R) \hookrightarrow M$ and a natural epimorphism $\rho: M^{**} \twoheadrightarrow \Ext^2(A(M),R)$ such that
\begin{equation} \label{epsilon} \tag{$\epsilon$}
  0 \to \Ext^1(A(M),R) \xrightarrow{\tau} M \xrightarrow{\epsilon} M^{**} \xrightarrow{\rho} \Ext^2(A(M),R) \to 0
\end{equation}
is an exact sequence.
In particular, $M$ is torsionless iff $\Ext^1(A(M),R)=0$ and reflexive iff $\Ext^i(A(M),R)=0$ for $i=1,2$.
A short elegant proof of \eqref{epsilon} can be found in \cite[Theorem~6]{CQR05} and a generalization in  \cite[Chapter~2, (2.1)]{AB} (see also \cite[Exer.~IV.7.3]{HS}).

A left (resp.~right) \textsc{Noetherian} domain $R$ satisfies the left (resp.~right) \textsc{Ore} condition and the set of torsion elements $\tor(M)$ of an $R$-module $M$ form an $R$-submodule.
The following theorem states that the two notions ``torsion-free'' and ``torsionless'' coincide for finitely generated modules.

\begin{theorem}[{\cite[Theorem~5]{CQR05}}] \label{thm:torsionless=torsion-free}
  Let $R$ be a two-sided \textsc{Noether}ian domain and $M$ a f.g.~$R$-module.
  Then the image of the natural monomorphism $\tau:\Ext^1(A(M),R) \to M$ is the torsion submodule $\tor(M)$ yielding a canonical isomorphism $\Ext^1(A(M),R) \cong \tor(M)$.
  In particular, the torsion-free factor and the torsionless factor of $M$ coincide and $M$ is torsion-free iff $M$ is torsionless.
\end{theorem}

\begin{coro}
  Let $R$ be a two-sided \textsc{Noether}ian domain.
  A finitely generated $R$-module of grade at least $1$ is torsion.
\end{coro}
\begin{proof}
  Let $T$ be a such a module.
  By Theorem~\ref{thm:torsionless=torsion-free} the torsion-free factor coincides with the torsionless factor.
  The latter is trivial since $\Hom_R(T,R)=0$ and the evaluation map $T \to T^{**}$ vanishes.
  Hence $T$ is a torsion module.
\end{proof}

Any finitely generated torsion-free module over a commutative domain can be embedded into a free module of finite rank.
This can be easily seen by passing to the quotient field (cf.~\cite[the paragraph preceding (2.31)]{Lam99}).
The exact sequence \eqref{epsilon} yields a generalization to the noncommutative case.
The following proposition is part of \cite[Theorem~8]{CQR05}.

\begin{prop} \label{prop:embed_in_free}
  Let $R$ be a two-sided \textsc{Noether}ian domain.
  A finitely generated torsionless (=torsion-free) $R$-module can be embedded in a free module of finite rank.
\end{prop}
\begin{proof}
  The two-sided coherence of $R$ assures the existence of finite rank free resolutions for f.g.~$R$-modules.
  Let $M \twoheadleftarrow F_0 \xleftarrow{\partial} F_1$ be a finite free presentation of $M$, i.e., with free modules $F_0$ and $F_1$ of finite rank.
  Dualizing we obtain a finite free presentation of $A(M)$ which we can resolve one step further and obtain $F_{-1}^* \to F_0^* \xrightarrow{\partial^*} F_1^* \to A(M)$ with $F_{-1}^*$ free of finite rank.
  Dualizing again yields an exact complex: The defect of exactness at $F_0^{**}$ is $\Ext^1(A(M),R)$ which vanishes since $M$ is torsionless.
  Using the reflexiveness of free modules of finite rank it follows that $M$ (as the cokernel of $\partial$ or $\partial^{**}$) embeds into the finite rank free module $F_{-1}^{**}$.
\end{proof}

The above Proposition is implemented for computable rings in \textsc{OreModules} \cite{CQR07} and $\mathtt{homalg}$ \cite{homalg-project,BL}.

\section{A module-theoretic generalization of Theorem~\ref{thm:diego}} \label{sec:module-theoretic_generalization2}

\begin{theorem}
  Let $R$ be a two-sided \textsc{Noether}ian domain and $M$ a finitely generated $R$-module.
  If $M$ is projective up to grade $2$ then the torsion-free factor $M/ \tor(M)$ is projective.
\end{theorem}
\begin{proof}
  Let  $P \stackrel{\pi}{\twoheadrightarrow} M$ be the f.g.~projective module of Definition~\ref{defn:projective_up_to_grade}, $A := \ker \pi \leq P$, $B$ be a complement up to the grade $2$ of $A$ in $P$, and $T := P / (A + B)$ the factor module of grade $\geq 2$.
  Let $A'$ denote the preimage of $\tor(M)$ in $P$, so $A'/A \cong \tor(M)$.
  The intersection $A' \cap B = 0$ since $A' \cap (A+B) = A$.
  The latter can be seen as follows:
  Otherwise $(A' \cap (A+B)) / A \leq A'/A \cong \tor(M)$ would be a nontrivial torsion\footnote{Here we need that $\tor(M)$ is torsion and not merely having a zero evaluation map.} submodule of the torsion-free factor $(A+B) / A \cong B$.
  The next proposition guarantees that the epimorphic image $T''=P / (A' + B)$ of $T$ is again of grade at least $2$.
  It remains to apply Theorem~\ref{coro:main1} to $M / \tor(M) \cong P / A'$ with $B$ now a complement of $A'$ in $P$ up to grade at least $2$.
\end{proof}

\begin{prop}
  Let $T$ be a torsion module  over a domain. If $T$ has grade at least $2$ then any of its factor modules has grade at least $2$.
\end{prop}
\begin{proof}
  The grade condition for $T$ means that $\Hom(T,R)=0=\Ext^1(T,R)$.
  Let $T'' = T/T'$ be a factor of $T$.
  Any morphism from a torsion module over a domain into a torsion-free module is zero.
  An since the submodule $T'$ is again torsion\footnote{Here we need that $T$ is torsion and not merely having a zero evaluation map.} it follows that $\Hom(T',R)=0$.
  The long exact $\Ext(-,R)$-sequence (w.r.t.~$0 \to T' \to T \to T'' \to 0$)
  \[
    0 \to \Hom(T'',R) \to \underbrace{\Hom(T,R)}_{=0} \to \underbrace{\Hom(T',R)}_{=0} \to \Ext^1(T'',R) \to \underbrace{\Ext^1(T,R)}_{=0}
  \]
  implies that $\Hom(T'',R) = 0 = \Ext^1(T'',R)$.
\end{proof}

We end this section by describing a context in which the original formulation can be retained.
If $M$ has a finite free resolution, e.g., if $R$ is an FFR ring\footnote{Finite free resolution ring.}, then, by a remark of \textsc{Serre}, $M$ projective implies $M$ stably free (cf.~\cite[Proposition~19.16]{eis}).
If, additionally, $R$ is \textsc{Hermite} then $M$ projective already implies $M$ free.
If $R$ is commutative \textsc{Cohen-Macaulay} then the notions of grade and codimension coincide (cf.~Remark~\ref{rmrk:CM}).
The rings mentioned in the Introduction are FFR, \textsc{Hermite}, and commutative \textsc{Cohen-Macaulay} (even regular) domains.

\begin{rmrk}
It should be noted that this paper is less of computational interest as non of the results suggests an algorithm to decide the projectivity of the torsion-free factor of a given finitely presented module.
For an overview on algorithms to test projectivity, stably freeness, and freeness see \cite[Subsection~3.4]{BL} and the references therein.
However, given $M=R^q/A$ and $B \leq R^q$ over a computable ring $R$ it can be algorithmically decided whether $A \cap B = 0$ and $\grade T \geq 2$ for $T=R^q/(A+B)$.
For the definition of a computable ring see \cite[Definition~3.2]{BL}.
The torsion-free factor over finitely presented modules over such rings can be computed, e.g., as the coimage of the evaluation map.
\end{rmrk}