Abstract
Abstract. It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that these in many cases can be obtained as inverse limits.
Introduction
Given a sequence (R n ) of rings with connecting morphisms (i.e. ring homomorphisms) π n : R n → R n−1 for all n (setting R 0 = R 1 and π 1 = id) we define the inverse limit as the ring lim ← − R n of strings x = (x n ) in R n , i.e. sequences such that π n (x n ) = x n−1 for all n. For each m there is a natural morphism ρ m : lim ← − R n → R m (the coordinate evaluation) obtained by evaluating a string x = (x n ) at m, and we see that π n • ρ n = ρ n−1 for every n. The ring R = lim ← − R n has the universal property that for each coherent sequence of morphisms σ n : S → R n from a ring S (i.e. π n •σ n = σ n−1 for all n) there is a unique morphism σ : S → R such that σ n = ρ n • σ for all n.
If for each m we let S m = ρ m (lim ← − R n ) ⊂ R m , then π n (S n ) = S n−1 and lim ← − R n = lim ← − S n . This shows that every ring which can be obtained as an inverse limit can also be obtained as an inverse limit in which each morphism π n is surjective. We shall refer to this case as a surjective inverse limit, and will concentrate exclusively on it.
By their very construction inverse limits tend to be large. For example, if the morphisms are not eventually constant, an inverse limit will be uncountable even if the rings are finite. This might be considered a detracting factor. In this paper we shall try to redeem the construction by showing that in many instances multiplier rings (which we expect to be large) can be obtained as inverse limits. Since the structural properties of inverse limits are good, we hereby obtain information about multiplier rings that would otherwise seem unreachable.
We prove in Section 1 that a surjective inverse limit of exchange rings is again an exchange ring. In Section 2 we show that the Bass stable rank of a surjective inverse limit of rings (R n ) is the supremum of the Bass stable ranks of the R n 's. In Section 3 we extend this to an important infinite case by proving that the surjective inverse limit of quasi-Bass rings is again a quasi-Bass ring.
In Sections 4 and 5 we establish the basic properties of approximate units and multiplier rings that we shall need. In particular we show that for every proper morphism ϕ : R → S between non-degenerate rings there is a unique extension ϕ : M(R) → M(S) between their multiplier rings. We also show that ϕ is strictly continuous, cf. [9] . If R is σ−unital, i.e. has a countable approximate unit, and ϕ is surjective we show that ϕ is surjective as well, thereby giving an algebraic analogue of the Tietze extension theorem (see e.g. [18] ).
Finally in Section 6 we combine our results by showing that many multiplier rings can be obtained as inverse limits. Thus if (R n ) is a sequence of σ−unital rings with surjective morphisms π n : R n → R n−1 we prove that M(lim ← − R n ) = lim ← − M(R n ). Going further we show that if a semi-prime σ−unital ring R has two sequences of ideals (I n ) and (J n ), one increasing to R, the other decreasing to 0, such that I n ∩ J n = 0 and R/J n is unital for every n, then M(R) = lim ← − R/J n . The inspiration for this paper comes from [15] , where a similar list of results are obtained for C * −algebras. In this category one must of course consider only bounded strings as elements in the inverse limit, so although our results are the same as in [15] , our proofs are completely different -sometimes harder, sometimes easier. It is worth mentioning that inverse limits of C * −algebras in the algebraic sense (with arbitrary, unbounded strings) have been explored by Phillips in [34] and [35] to obtain examples of pro C * −algebras. The aim was to create a non-commutative analogue of normal spaces that are not necessarily locally compact. The unpublished notes [20] by Goodearl have also influenced our work, although we fail to answer the question that motivated them: Will a surjective inverse limit of separative exchange rings itself be separative? A negative answer would provide a solution to the Fundamental Separativity Problem (see [4] ).
Exchange Rings
Recall from [21] and [28] that a (unital) ring R is an exchange ring if for each x in R there is an idempotent e in xR such that 1 − x = (1 − e)(1 − y) for some y in R. This is not the original definition, which concerns a finite exchange property for R−modules, see [39] , but an equivalent formulation better suited for our purposes.
The class of exchange rings is pleasantly large and includes all (von Neumann) regular rings, all π−regular rings, the semi-perfect rings (identified with the semi-local exchange rings) and the C * −algebras of real rank zero (identified with the exchange C * −algebras by [4, Theorem 7.2]).
We shall show that every surjective inverse limit of exchange rings again has the exchange property. As an hors d'oeuvre we present the following simpler result from [20] (see also [19, Example 1.10] ). Recall that an element x in a ring R is said to be regular provided that x = xyx for some y in R. We shall refer to such an element y as a partial inverse for x. If all elements of R are regular we say that R is (von Neumann) regular. Proposition 1.1. (Goodearl) If R is the surjective inverse limit of a sequence (R n ) of regular rings, then R is also regular.
Proof. Put R = lim ← − R n and let π n : R n → R n−1 denote the connecting morphisms. If x = (x n ) is an element in R and if for all k < n we have found elements y k in R k such that π k (y k ) = y k−1 and x k y k x k = x k , we choose an element y in R n with π n (y) = y n−1 . Then u = x n − x n yx n ∈ ker π n , and since this ideal is a regular ring in its own right we can find a v in ker π n such that uvu = u. By computation x n − x n yx n = (x n − x n yx n )v(x n − x n yx n ) = x n vx n − x n vx n yx n − x n yx n vx n + x n yx n vx n yx n , and it follows that the element y n = y + v − yx n v − vx n y + yx n vx n y n is a partial inverse for x n with π n (y n ) = y n−1 .
By induction we can therefore find an element y = (y n ) in R such that x = xyx, so every element in R is regular, as desired.
Note that in the definition of an exchange ring the role of the unit is superfluous. Thus we say that a (not necessarily unital) ring R is an exchange ring if for each x in R there is an idempotent e in xR such that x = e + y − ey for some y in R. This idea was successfully exploited by Ara in [1] . If I is a non-unital exchange ring embedded as a two-sided ideal of a unital ring R, we will adopt the terminology from [1] and say that I is an exchange ideal of R. Lemma 1.2. Let I be a two-sided exchange ideal in a unital ring R. If x and y are elements in R and p is an idempotent such that p − xy ∈ I, there is an idempotent q in I and an element r in pRp with p − r in I, such that both elements a = (p − q)x and b = yr are regular and partial inverses for one another. By construction a−px ∈ I and b−yp ∈ I, so ab−p ∈ I. This is a restatement of [1, Lemma 2.1]. The result is obtained by applying the exchange condition to the element p − pxyp in pIp to get q.
If R is a non-unital ring, it is sometimes convenient to adjoin a unit in order to obtain the ring R + = R⊕Z, with elementwise addition and multiplication given by the rule (x, n)(y, m) = (xy + mx + ny, nm). In this way R sits as a two-sided ideal in R + with quotient Z. Lemma 1.3. Let π : R → S be a surjective morphism between (not necessarily unital) exchange rings, and let x, y and z be elements in S such that with e = x y we have
For each choice of x in R with π(x) = x there are then elements y and z in R with π(y) = y and π(z) = z, such that with e = xy we have
Proof. This proof amalgamates arguments from [1, Theorem 2.2] and [28, Proposition 1.1], which are included for convenience. As usual we shall write p ≤ q for idempotents p and q such that pq = qp = p. Moreover, if R is not unital we adjoin a unit to obtain the unital ring R + = R ⊕ Z and we identify S with an exchange ideal in S + , setting π(1) = 1. Observe first that e is an idempotent by necessity, since e e = x y e = x y = e. Thus also p 1 = y x is an idempotent. Since R is an exchange ring p 1 can be lifted to an idempotent p 1 in R by [1, Theorem 2.2] . Let y 1 and z 1 be any lifts in R of the elements y and z. By Lemma 1.2 (with I = ker(π) and exchanging the roles of x and y) we can find regular elements a 1 and b 1 in R that are partial inverses for one another, such that a 1 = (p 1 − q 1 )y 1 and b 1 = xr 1 for some idempotent q 1 in I and some r 1 in p 1 Rp 1 with π(r 1 ) = p 1 . Thus e 1 = b 1 a 1 is an idempotent with π(e 1 ) = x p 1 p 1 y = e e e = e .
is an idempotent in 1 − S and lift it to an idempotent p 2 in 1 − R. Again by Lemma 1.2 there are regular elements a 2 and b 2 in 1 − R, partial inverses for one another, such that a 2 = (p 2 − q 2 )(1 − z 2 ) and b 2 = (1 − x)r 2 for some idempotent q 2 in I and some element r 2 in p 2 R + p 2 with π(r 2 ) = p 2 = (1 − z)(1 − x) (so that r 2 ∈ 1 − R as well). The idempotent f = b 2 a 2 belongs to (1 − x)(1 − R)(1 − z 2 ) and satisfies
Since f = f (1 − e 1 ) by construction of z 2 , we may consider the idempotent (1 − e 1 )f , where
Now apply the exchange property to the element qxq in the corner ring qRq = qR + q (⊂ I) to find an idempotent t in qRq such that t = qxqcq and q − t = q(1 − x)qdq for some elements c, d in R. Put s = xqct, and note that sq = s and qs = t, whence s 2 = s. Since t ≤ q ≤ 1 − e 1 , the element e 2 = e 1 + (1 − e 1 )s is an idempotent in R with e 1 ≤ e 2 and π(e 2 ) = e. Let y 2 = r 1 a 1 (1 − s) + qct. Then π(y 2 ) = p 1 y = y x y = y. Moreover,
The argument in [1, Theorem 2.2] proceeds to show that the right ideal A = e 2 R + + (1 − x)R + equals R + , and does so by showing that both 1 − q and q belong to A. All we need is the last assertion, but there seems no easy way to obtain the specific decomposition of q except by the full argument.
Evidently e 2 e 1 ∈ A and e 2 (1 − e 1 ) ∈ A. We know that e 2 e 1 = e 1 , and therefore e 2 (1 − e 1 ) = e 2 − e 1 = (1 − e 1 )s = s − e 1 s. Consequently both e 1 ∈ A and s ∈ A. Now f = (1 − x)r 2 a 2 ∈ A, so also (1 − e 1 )f = f − e 1 f ∈ A. It follows that 1 − q = e 1 + (1 − e 1 )f ∈ A. As s ∈ A and t = qs we now conclude that t = s − (1 − q)s ∈ A. Since
we finally see that q = q − t + t ∈ A. We can therefore write
for some elements u and v in Rq. In particular, both elements belong to I. Put z 3 = 1 − (r 2 a 2 + v) and e 3 = e 1 (1 − f ) + u, and note that π(1 − z 3 ) = p 2 p 2 (1 − z) = (1 − z)(1 − x)(1 − z) = 1 − z and π(e 3 ) = e e = e. Moreover, by (2)
We can now make our final choices as follows:
It is easy to check that
Moreover, by (1) and (3) we have the desired relations: xy = xy 2 e = e 2 e = e ye = y and (1 − z)e = 0
= (1 − e 2 − e 2 e 3 (1 − e 2 ))(1 − e) = (1 − e)(1 − e) = 1 − e . Theorem 1.4. If R is the surjective inverse limit of a sequence (R n ) of exchange rings, then R is also an exchange ring.
Proof. Consider an element x in R, identified with a string (x n ) in R n . We must then find an idempotent e = (e n ) in R such that e ∈ xR and 1 − e ∈ (1 − x)(1 − R). Since R 1 is an exchange ring we can find an idempotent e 1 and elements y 1 , z 1 such that e 1 = x 1 y 1 and 1 − e 1 = (1 − x 1 )(1 − z 1 ). Evidently we may also assume that y 1 = y 1 e 1 and (1 − z 1 )e 1 = 0.
Assume now that for some n we have found elements y k and z k in R k for 0 ≤ k ≤ n, such that π k (y k ) = y k−1 and π k (z k ) = z k−1 for all k ≥ 2, and moreover with e k = x k y k we have the relations
, for all k. By Lemma 1.3 we can then find elements y n+1 and z n+1 in R n+1 with π n+1 (y n+1 ) = y n and π n+1 (z n+1 ) = z n , such that with e n+1 = x n+1 y n+1 we have
By induction this defines elements y = (y n ) and z = (z n ) in R, such that the element e = xy is an idempotent and 1 − e = (1 − x)(1 − z), as desired.
Bass Stable Rank
Let R be a unital ring. As usual we say that a row a = (a 1 , .
To facilitate the computations with rows in
and s ∈ R, we write sa = (sa 1 , . . . , sa d ).
Recall from [10] (see also e.g. [27] , [25] ) that the Bass stable rank of a unital ring R is the smallest number bsr(R) such that for each d ≥ bsr(R) every right unimodular row a = (a 0 , . . . , a d ) in R d+1 can be reduced to a right unimodular row in R d of the form a r + a 0 b for a suitable row b in R d . Here a r = (a 1 , . . . , a n ) and we regard R d as a two-sided module over R. Evidently this definition favours right unimodular rows and should be called the right Bass rank, but it turns out that the analogous concept for left unimodular rows gives the same lower bound ( [37] , [27] ). In particular bsr(R) = 1 (in which case we say that R is a Bass ring) if every equation ax + b = 1 implies that a + by is invertible in R for a suitable y, cf. [38] . 
Proof. Since bsr(R) ≤ d, the equation a · x + s1 = 1 produces rows b and y in R d such that (a + sb) · y = 1. Define the elements y = a · y, b = b · y and z = 1 − b · x, and note that y + sb = 1. By computation we therefore get Proof. Take any lifts b
It follows from (4) that with s = s · (y ′ + t) in I we have
Using Lemma 2.1 on (5) we find rows u in R d and v in (Ra 0 bsR)
Clearly the rows b = b ′ + bsu and y = y ′ + t + v verify the desired conditions.
Proof. Let π n : R n → R n−1 with R 0 = R 1 and π 1 = id, and consider an equation a · x = 1 in R d+1 . Identify the rows a and x with strings (a n ) and (x n ) in R d+1 n . Write a n = (a n,0 , . . . , a n,d ). Since bsr(R 1 ) ≤ d there are rows b 1 and y 1 in R 
Proof. For each n the coordinate projections ρ n : R → R n are surjective, whence bsr(R n ) ≤ bsr(R) (see, e.g. [37, Theorem 4] ). Thus sup n bsr(R n ) ≤ bsr(R).
For the converse inequality, we may of course assume that bsr(R n ) ≤ d for all n (and some finite d). By Theorem 2.3 bsr(R) ≤ d as well.
QB−Rings
As shown in [6] a QB−ring is an infinite version of a Bass ring, i.e. a ring R with bsr(R) = 1. To arrive at the definition we replace the set R −1 of invertible elements in R with the set R
(We then write 1 − vu ⊥ 1 − uw.) It follows easily that u is a (von Neumann) regular element in R, and that one may take v = w and demand that u and v are partial inverses for one another (i.e. u = uvu and v = vuv). We refer to this situation by saying that v is a quasi-inverse for u. The ring R is then a quasi-Bass ring (a QB−ring for short) if whenever ax + b = 1 in R we can find y in R such that a + by ∈ R −1 q , cf. the definition of Bass rings in the introduction to Section 2. As with the notion of stable rank, the concept of QB−rings is left-right symmetric, see [6, Theorem 3.6] .
The theory of QB−rings has been developed in the papers [6] , [7] and [8] with the aim of extending as much as possible of the theory of Bass rings. The inspiration is the corresponding series of papers [12] , [13] , [14] , [15] and [16] , in which the theory of C * −algebras of topological stable rank one is being extended to the class of extremally rich C * −algebras, which are the C * −analogues of QB−rings.
We are going to prove that any surjective inverse limit of QB−rings is again a QB−ring. To establish this in full detail turns out to be quite intricate, and we would have preferred an easier, more accessible proof. The basic techniques come from [8, Lemmas 1.4 and 1.5], and we include part of the discussion carried out there. The following Proposition examines the situation in an easier setting, hence gives an idea of what is going on.
In Theorem 3.3 below we are going to use the left-handed version of the concept of a QB−ring. This is done in order to make the techniques developed in [8] readily accessible. Although it will not be strictly necessary for Proposition 3.1 we have chosen the same version there to avoid confusion.
For the convenience of the reader we are only going to prove the unital versions of these results. The non-unital versions follow by straightforward, but sometimes quite exasperating computations, where invertible and quasi-invertible elements are replaced by adversible and quasi-adversible elementsà la Kaplansky, cf. [6, Section 4] . Basically an element u in a nonunital ring R is (left/right/quasi) adversible if 1 − u is (left/right/quasi) invertible in some (hence any) unital ring R containing R as an ideal.
Recall that a non-unital ring I has stable rank one if whenever (1 − x)(1 − a) + b = 1 in I + , where x, a ∈ I, we can find y in I such that 1 − a + yb ∈ (I + ) −1 .
Proposition 3.1. Let (R n ) be a sequence of unital QB−rings, and assume that we have surjective morphisms π n : R n → R n−1 such that ker π n has stable rank one for every n. Then R = lim ← − R n is also a QB−ring.
Proof. Arguing along the lines of Theorem 1.4 and Theorem 2.3, it suffices to consider the case of a surjective morphism π : R → S, where R and S are unital QB−rings and I = ker π has stable rank one, along with equations xa + b = 1 in R and x a + b = 1 in S, so that π(a) = a, π(b) = b and π(x) = x. Since S is a QB−ring, there are elements y in S and u in S −1 q such that a + yb = u. We wish to find elements y in R and u in R −1 q such that π(y) = y, π(u) = u and a + yb = u.
First, by [6, Proposition 7.1] lift u to a quasi-invertible element u 1 in R, and let y 1 be any lift of y. We then get
where t ∈ ker π = I. Next, choose a quasi-inverse v 1 for u 1 . By computation
so, multiplying left and right with u 1 and v 1 we have
Rearranging terms this gives the equation
By [6, Lemma 4.6] we can choose t 1 and s 1 both in I such that
Since I has stable rank one we can find s 2 in I such that
and we note that 1 − w ∈ I. It follows that we can write
we have that the element
Let R be a unital ring with an ideal I. Consider an element u in R −1 q with quasi-inverse v, so that u = uvu and v = vuv, and if p = uv and q = vu then (1 − p) ⊥ (1 − q). Let w be an element in (qRq) Proof. As in Proposition 3.1 we only need to consider the case of a surjective morphism π : R → S, where R and S are unital QB−rings, along with equations xa + b = 1 in R and x a + b = 1 in S, so that π(a) = a, π(b) = b, π(x) = x. Since S is a QB−ring, there are elements y in S and u in S −1 q such that a + yb = u. We wish to find elements y in R and u in R −1 q such that π(y) = y, π(u) = u and a + yb = u. Lift u to a quasi-invertible element u 1 in R (again by [6, Proposition 7.1]), and let y 1 be any lift of y. We then get a + y 1 b = u 1 + t , where t ∈ ker π = I. By computation,
The computations carried out in [8, Lemma 1.5] can be performed in this situation also (to the equation (6)). Hence we obtain an element w in (qRq) −1 q ∩ (q + qIq) and an element z in I such that, if w ′ is any quasi-inverse for w, and if we let
where w 1 , w 2 are invertible elements in R, and a 1 = u 1 w + t 1 + t 2 , with t i in p i Rq i for i = 1, 2.
Since x(u 1 + t) + b 1 = 1, we also have
Conjugating with
as desired.
Approximate Units
σ−Unital Rings. Contrary to popular belief rings do not come automatically equipped with a unit. Of course, a unit can be adjoined, passing from R to R + as decribed before, but his may destroy other desirable relations for the ring, like being an ideal in a larger ring or being a Bass stable ring or a QB−ring. Also, indiscriminate adjoining of units may deprive the category of rings some of the applications of general category theory.
For some rings adjoining of a unit may be unneccesary because the ring itself already possesses elements that locally act as units. As in [9] a net (e λ ) λ∈Λ in R is called an approximate unit for R if eventually e λ x = xe λ = x for every x in R. We do not expect that the net consists of idempotents, but in many cases it will (for example, in the case where R is an exchange ring). Far more important is the case where we can choose Λ = N. We say in this case that R is σ−unital. Evidently we may then choose the approximate unit (e n ) such that e n+1 e n = e n e n+1 = e n for all n, see [9, Lemma 1.5]. Note that for such a sequence to be an approximate unit it suffices to show that for each x in R there is some n such that e n x = xe n = x. The same equations will then automatically hold for all indices larger than n. Thus in what follows we shall often assume that a σ−unit (e n ) satisfies e n e n+1 = e n+1 e n = e n for all n.
Rings with approximate units are examples of s−unital rings (see e.g. [36] ). If the ring is countable it is σ−unital if and only if it is s−unital, see [2, Lemma 2.2].
It should also be remarked that all C * −algebras and all classical (non-unital) Banach algebras have approximate units in the topological sense, which implies that certain dense ideals -representing elements with compact support or finite rank -have approximate units in the algebraic sense. It is instructing though to note that the ring B f (H) of operators with finite rank on an infinite dimensional Hilbert space H does not have a countable approximate unit (although it certainly has an (uncountable) approximate unit consisting of projections). By contrast, the ring M ∞ (R) = lim − → M n (R) of finite matrices over a unital or just σ−unital ring R is σ−unital. Proof. Let π : R → S denote the quotient map and choose an approximate unit (e n ) for I. Choose also a sequence (f n ) in R such that (π(f n )) is an approximate unit for S.
For each x in R there is an n such that π(f n x) = π(xf n ) = π(x). Thus (1 − f n )x ∈ I and x(1 − f n ) ∈ I, so for a suitable m we have (1 − e m )(1 − f n )x = 0 and x(1 − f n )(1 − e m ) = 0. It follows that if we define u nm in R by
(Note here that the use of 1 is purely formal.) We claim that for each (p, q) in N 2 there is an n > p and m > q such that
for all k ≤ p and l ≤ q. This is equivalent to the claim that
together with the similar adjoint version. Note now that (1 − f k )(1 − e l )(1 − f k ) = 1 − a kl for some elements a kl in R. Choosing n sufficiently large we may assume that (1 − f n )a kl ∈ I for all k ≤ p and l ≤ q. For a sufficiently large m we therefore have (1 − e m )(1 − f n )a kl = 0 for all k ≤ p and l ≤ q. Consequently,
as desired. The proof for the adjoint version is quite similar. Using the claim we now inductively choose a sequence (v i ) in R with v i = u n(i)m(i) such that v i u nm = u nm v i = u nm for all n ≤ i and m ≤ i; but also such that v i+1 v i = v i v i+1 = v i for all i. As we saw above, there is for each x in R some (n, m) in N 2 such that u nm x = xu nm = x, and it follows that if i ≥ max{n, m} then
and similarly xv i = x. Thus (v i ) is a σ−unit for R. Definition 4.2. If I is an ideal in a ring R we say that I has a unit in R if ex = xe = x for some e in R and every x in I. Similarly we say that I has a σ−unit in R if there is a sequence e n in R such that eventually e n x = xe n = x for each x in I.
Inspection of the proof of Proposition 4.1 shows that in order for R to be σ−unital it suffices that S is σ−unital and I has a σ−unit in R. In this formulation the requirements are both necessary and sufficient.
The argument also shows that R is unital if and only if S is unital and I has a unit in R. If e is a unit for I in R and f is an element in R such that π(f ) is the unit in S then the element u in R given by 1 − u = (1 − f )(1 − e)(1 − f ) is the unit in R. Proof. If R is σ−unital then so is every R n , since the morphisms ρ n are surjective. The ideals ker ρ n form a decreasing sequence, so if they do not eventually have a unit in R none of them will. To obtain a contradiction, let us assume this to be the case, and let (e n ) be an approximate unit for R.
Given x 1 in R 1 we claim that there is an x in R such that ρ 1 (x) = x 1 and either e 2 x = x or xe 2 = x. For if this were not the case then e 2 (x + s) = (x + s)e 2 = x + s for each fixed x in R with ρ 1 (x) = x 1 and every s in ker ρ 1 . In particular, e 2 would be a unit for ker ρ 1 in R, contradicting our assumptions. Assuming therefore, as we may, that e 2 x = x we can find a (first) coordinate k 2 in which they differ. Thus, if we let
By the same argument we can now find an element y in R with ρ k 2 (y) = x 2 , such that either e 3 y = y or ye 3 = y. This produces a coordinate number k 3 and an element x 3 = ρ k 3 (x) such that either ρ k 3 (e 3 )x 3 = x 3 or x 3 ρ k 3 (e 3 ) = x 3 and
Continuing by induction we find a coherent sequence x = (x n ) in (R kn ), i.e. π kn−1 • · · · • π k n+1 (x n+1 ) = x n for all n, such that for each n either ρ kn (e n )x n = x n or x n ρ kn (e n ) = x n . Identifying x = (x n ) with an element in R we see that either e n x = x or xe n = x for every n, contradicting the assumption that (e n ) was an approximate unit.
In the converse direction, if R k is σ−unital and ker ρ k has a unit in R, then R is σ−unital by Proposition 4.1 and the comments after Definition 4.2 applied to the extension
The Theorem above shows that in all but a few cases a surjective inverse limit of non-unital rings will not be σ−unital. The demand of an eventual unit for ker ρ n in R translates to the demand that with the morphisms π m : R m → R m−1 there is a unit e m for ker σ m in R m for each m > n, where σ m = π n+1 • π n+2 • · · · • π m , and moreover these can be chosen coherent, i.e. π m+1 (e m+1 ) = e m for all m. Clearly this is not likely to happen in actual examples.
Multiplier Rings
Centralizers. Throughout this section we assume that R is a non-degenerate ring, i.e. xR = 0 or Rx = 0 implies x = 0 for every x in R. This concept is of course only relevant for non-unital rings, but note that rings with approximate units will automatically be nondegenerate. Eventually we are going to need the stronger concept that R is semi-prime, which means that xRx = 0 implies x = 0 for every x in R.
Following Hochschild, [22, Definition 3.1], although using the terminology from [23] , we define a left centralizer to be a map λ : R → R such that λ(xy) = λ(x)y for all x, y in R. Similarly, a right centralizer ρ is a map such that ρ(xy) = xρ(y). A double centralizer is a pair (λ, ρ) of left-right centralizers satisfying the coherence relation xλ(y) = ρ(x)y for all x, y in R. It is easy to see that (in the presence of non-degeneracy) this relation alone will imply that λ and ρ are both R−module maps, cf. [23, Theorem 7] . For such pairs we define a product by setting (λ 1 , ρ 1 )(λ 2 , ρ 2 ) = (λ 1 • λ 2 , ρ 2 • ρ 1 ) . With the obvious sum (λ 1 , ρ 1 )+(λ 2 , ρ 2 ) = (λ 1 +λ 2 , ρ 1 +ρ 2 ) the set M(R) of double centralizers becomes a unital ring with the unit 1 = (id, id). We shall refer to M(R) as the multiplier ring for R.
For each x in R we define the left and right centralizers λ x and ρ x by λ x (y) = xy and ρ x (y) = yx. Then (λ x , ρ x ) is a double centralizer, and since (λ x , ρ x )(λ y , ρ y ) = (λ xy , ρ xy ) it follows that we have a morphism x → (λ x , ρ x ) of R into M(R). The non-degeneracy of R ensures that this morphism is injective, so we may identify R with its image in M(R). The easily verified formulas
show that
which proves that R is a two-sided ideal in M(R). At the same time we see from (7) that R is an essential ideal in M(R), because R(λ, ρ) = 0 or (λ, ρ)R = 0 for some (λ, ρ) in M(R) implies that λ = 0 or ρ = 0, which forces (λ, ρ) = 0 by the coherence relation. It is easy to show that the multiplier ring has the universal property that if S is any ring containing R as an ideal there is a morphism ϕ : S → M(R) extending the embedding morphism R → S. Moreover, ϕ is injective precisely when R is an essential ideal in S. For each y in S one just defines ϕ(y) = (λ y , ρ y ). The multiplier ring M(R), or rather its quotient M(R)/R, is an indispensable tool in Hochschild's classification of extensions of R, see the remarks at the end of the paper. It also occurs naturally as the function-theoretic method to describe the Stone-Čech compactification of a (locally compact) topological space. In K−theory it is used to describe the ring B(R) of row-and column-finite matrices over a unital ring R. Specifically, if M ∞ (R) = lim − → M n (R), identified with the ring of finite matrices over R, we find that its multiplier ring is identified with B(R) (see e.g. [9, Proposition 1.1]). Definition 5.1. A morphism π : R → S between rings R and S is called proper of π(R) is not contained in any proper left or right ideal of S, i.e. Sπ(R) = π(R)S = S (in the sense that π(R)S denotes the set of finite sums of products π(x)y). If R is unital then π is proper if and only if both S and π are also unital.
The origin of this notion comes from C * −algebra theory, cf. [17] or [26] , where one notices that if X and Y are locally compact Hausdorff spaces and R = C 0 (X) and S = C 0 (Y ) denote the rings of complex-valued continuous functions vanishing at infinity, then a morphism (which in this category means a *-homomorphism) π : R → S is proper if and only if it is the transposed of a proper continuous map f : Y → X, i.e. one for which f −1 (C) is compact in Y for every compact subset C in X.
Theorem 5.2. Let π : R → S be a proper morphism between non-degenerate rings R and S.
There is then a unique unital morphism π : M(R) → M(S) that extends π.
Proof. If (λ, ρ) ∈ M(R) and y, z are elements in S we can write them in the form y = π(u i )y i and z = z j π(v j ) for suitable finite subsets {u i }, {v j } in R and
Similarly, ρ(z)π(x) = zπ(λ(x)). For one thing, this implies that the elements λ(y) and ρ(z) are independent of the representations of y and z. If namely λ ′ (y) arises from another representation y = π(u
)(y − y) = 0 for all x in R, which forces λ(y) = λ ′ (y) by properness and non-degeneracy. On the other hand the equations show that we have the coherence relation
We may therefore define π(λ, ρ) = (λ, ρ) in M(S), and it is straightforward to check that π is a unital morphism from M(R) to M(S). By construction π is an extension of π, and since S is an essential ideal of M(S) there can be only one extension, so π is unique.
The Strict Topology. If R is a non-degenerate ring with multiplier ring M(R) we can for each a in R define O a = {x ∈ M(R) | xa = ax = 0}. As shown in [9] the sets O a form a subbasis for the neighbourhood system around 0 in the strict topology on M(R). In this topology addition is continuous (by construction) and multiplication is continuous by [9, Lemma 1.3] . A strict Cauchy net (x λ ) in M(R) is a net such that each product net (ax λ ) or (x λ a) is eventually constant. It follows easily from this that M(R) is strictly complete, cf. [9, Proposition 1.6].
If R has an approximate unit (e λ ), then e λ → 1 strictly in M(R), and R is therefore strictly dense in M(R). Conversely, if 1 is a strict limit point for R then every net (e λ ) in R converging strictly to 1 will be an approximate unit for R, and R is strictly dense in M(R), cf. [9, Proposition 1.6]. Proof. Let (x λ ) be a net in M(R) converging strictly to 0. Given y in S we can then by properness write y = π(x i )y i . Consequently
eventually. Similarly yπ(x λ ) = 0 eventually, which proves that π(x λ ) → 0 strictly in M(S).
The next theorem is the algebraic counterpart of a rather useful result from C * −algebra theory, known as the non-commutative Tietze extension theorem, cf. [31, Proposition 3.12.10] or [32, Theorem 10] . Applied to the ring R = C 0 (X) of complex-valued continuous functions on some locally compact Hausdorff space X, we find that M(R) = C b (X), the ring of bounded continuous functions on X. The content of the theorem is then that for any bounded continuous function f on a closed subset Y of X (and these are the only ways quotients of the ring C 0 (X) can arise), there is an extension f of f to a bounded continuous function on all of X. The necessity of a countable approximate unit for R reflects the fact that a general locally compact space X need not be normal (and only for normal spaces will the Tietze extension theorem hold), but if in addition X is σ−compact, then it is normal. Proof. Let (e n ) be a countable approximate unit for R, and without loss of generality assume that e n+1 e n = e n = e n e n+1 for all n. Consider now an element x in M(S). Since the elements e n = π(e n ) form an approximate unit for S we have for each n that e m x e n = x e n for m large enough. Passing to a subsequence we may therefore assume that (8) e n+1 x e n = x e n for all n.
Assume that for some n ≥ 2 we have found elements x 1 , . . . , x n in R with π(x k ) = e k x such that with e −1 = e 0 = 0 we have
for 2 ≤ k ≤ n. Choose z in R with π(z) = e n+1 x. Then by (8) π((z − x n )e n−1 ) = (e n+1 − e n )x e n−1 = 0 , π(e n−1 (z − x n )) = e n−1 (e n+1 − e n )x = 0 .
Thus (z − x n )e n−1 ∈ ker π and e n−1 (z − x n ) ∈ ker π. Now define
so that π(x n+1 ) = e n+1 x. Moreover,
This means that e n−2 (x n+1 − x n ) = 0 = (x n+1 − x n )e n−2 . By induction we can therefore find a sequence (x n ) in R satisfying (9) such that also π(x n ) = e n x for all n ≥ 3.
For each a in R there is a number k such that e k a = a = ae k . It follows from (9) that (x m − x n )a = (x m − x n )e k a = 0 , and similarly a(x m − x n ) = 0 for all n, m ≥ k+2. Consequently (x n ) is a strict Cauchy sequence in M(R), hence convergent to an element x in M(R). Since π is strictly continuous by Proposition 5.3 we finally see that
as desired.
Multiplier Rings as Inverse Limits
As before we let R = lim ← − R n for a sequence of rings R n with surjective morphisms π n : R n → R n−1 and consider the surjective coordinate evaluation morphisms ρ n : R → R n . If I is an ideal in R we evidently obtain a sequence (I n ) of ideals, where I n = ρ n (R) ⊂ R n , and by restriction this defines surjective morphisms π n : I n → I n−1 . Conversely, if we have a sequence of ideals I n ⊂ R n such that π n (I n ) = I n−1 for all n, we may identify the inverse limit lim ← − I n with an ideal I in R. Note that if we start with an ideal I in R then I ⊂ I, and in general the inclusion is strict.
Theorem 6.1. If lim ← − R n is the surjective inverse limit of a sequence of σ−unital rings and I is an ideal of lim ← − R n such that ρ n (I) = R n for every n then
Proof. Put R = lim ← − R n and M = lim ← − M(R n ). We then claim that there is a commutative diagram I
Here ι and ι k for k ≥ 0 are the natural embeddings, and π n is the surjective morphism obtained from Theorem 5.4. It follows that M is the surjective inverse limit of the multiplier rings M(R n ), and the coordinate evaluations ρ n are therefore also surjective.
Since the right-hand square of the diagram is commutative we can define the morphism ρ by (x n ) → (ι n (x n )) for every string x = (x n ) in R, and we note that ρ n • ρ = ι n • ρ n by this definition. Evidently ρ is injective. If moreover y = (y n ) is a string in M then for any x in R we have that ρ n (ρ(x)y) = ι n (x n )y n = x n y n . Identifying (x n y n ) with a string in R (using the commutativity of the diagram) we see that ρ(R) is an ideal in M, which must even be essential, since ρ(R)y = 0 implies R n ρ n (y) = ρ n (R)ρ n (y) = ρ n (ρ(R)y) = 0, hence ρ n (y) = 0 for all n by non-degeneracy, and so y = 0.
We claim that I is essential in R. For if xI = 0 for some x in R, then ρ n (x)R n = 0 for every n by our assumption on I, whence ρ n (x) = 0 since R n is non-degenerate (being σ−unital), and therefore x = 0. Since the property of being essential is hereditary it follows that ρ(ι 0 (I)) is an essential ideal in M. By the universal property of multiplier rings there is therefore an injective morphism ϕ :
Each surjective morphism ρ n •ι 0 extends uniquely to a (not necessarily surjective) morphism ψ n : M(I) → M(R n ) by Theorem 5.2. Since π n • ρ n • ι 0 = ρ n−1 • ι 0 and I is essential in M(I) it follows that also π n • ψ n = ψ n−1 for all n. By the universal property of inverse limits this means that we have a unique morphism ψ : M(I) → M such that ρ n • ψ = ψ n for all n. It follows that
Combining these results we find that
Since ι(I) is an essential ideal in M(I) and ρ(ι 0 (I)) is an essential ideal in M these equations imply that ϕ and ψ are the inverse of one another, and we have our natural isomorphism.
is a sequence of σ−unital rings with surjective morphisms π n : R n → R n−1 , and if π n : M(R n ) → M(R n−1 ) denote the unique surjective extensions of the π n 's then
O'Meara proved in [29] that if R is a σ−unital von Neumann regular ring then M(R) is an exchange ring (but is generally not regular). His arguments used critically the σ−unital assumption on the ring R. In the next result we show that in certain cases, multiplier rings of not necessarily σ−unital regular rings are also exchange rings. These are obtained as inverse limits of σ−unital regular rings, which will generally not be σ−unital themselves, cf. Theorem 4.3. Let as before R = lim ← − R n be a surjective inverse limit of a sequence of σ−unital rings. Assume moreover that all R n are semi-prime. This implies that also R is semi-prime. For if xRx = 0 for some x in R then ρ n (x)R n ρ n (x) = 0 for every n, whence ρ n (x) = 0 so that x = 0. If now I is an ideal in R we define the (two-sided) annihilator ideal by Definition 6.4. Let lim ← − R n be a surjective inverse limit of a sequence of semi-prime rings. We say that an ideal I in lim ← − R n is m−constant if I ∩ ker ρ m = 0. Equivalently, I ⊂ (ker ρ m ) ⊥ . We may graphically view ker ρ m as the strings x = (x n ) in R such that x n = 0 for n ≤ m. Evidently these coordinate ideals decrease, so that if I is m−constant, then I ∩ ker ρ n = 0 for all n ≥ m.
Since π n • ρ n = ρ n−1 we see that ker π n ⊂ ρ n (ker ρ m ) for n > m. If therefore I n = ρ n (I) denotes the associated sequence of ideals in R n , then I n ∩ ker π n = 0 for n > m. Thus I is isomorphic to I m and I n is isomorphic to I m for all n ≥ m. In particular, lim ← − I n = I. Conversely, if (I n ) is a sequence of ideals in (R n ) such that π n (I n ) = I n−1 for all n and I n ∩ ker π n = 0 for all n ≥ m for some m then I = lim ← − I n will be an m−constant ideal in lim ← − R n . If I is an n−constant and J is an m−constant ideal with n ≤ m, then I+J ⊂ (ker ρ m )
⊥ since ker ρ m ⊂ ker ρ n , so I + J is an m−constant ideal. Thus (ker ρ m )
⊥ is the largest m−constant ideal. It follows that I c = (ker ρ m )
⊥ is equal to the sum of all constant ideals, and we shall refer to it as the quasi-constant ideal of R.
The motivating example for considering constant and quasi-constant ideals arises from the Stone-Čech compactification. If X is a locally compact Hausdorff space then its StoneCech compactification βX has the property that C(βX) = C b (X) = M(C 0 (X)), and this ring is always a C * −inverse limit. In the case where X is also σ−compact we can write X = X n , where each X n is a compact subset of X contained in the interior X • n+1 of X n+1 . Put R n = C(X n ) and let π n (f ) = f |X n−1 for each f in C(X n ). Then C b (X) = lim ← − R n . The large constant ideals will be of the form (ker π m )
, so the quasi-constant ideal of C b (X) can (in this category) be identified with the norm-closed ideal C 0 (X).
If the construction is carried out in a purely algebraic context we find that the ring C(X) of all continuous functions on X is the (algebraic) inverse limit of the sequence (C(X n )), and the quasi-constant ideal of C(X) is the ring C c (X) of continuous functions with compact supports.
Theorem 6.1 provides an immediate generalization of this construction:
Corollary 6.5. If R = lim ← − R n is the surjective inverse limit of a sequence of σ−unital rings such that the quasi-constant ideal I c of R satisfies ρ n (I c ) = R n for every n, then M(I c ) = lim ← − M(R n ).
Theorem 6.6. Let (I n ) and (J n ) be two sequences of ideals in a semi-prime σ−unital ring R, one increasing, the other decreasing, but such that I n ∩ J n = 0 for all n. If I = I n is essential in R and I + J n = R for every n, then with R n = R/J n and π n : R n → R n−1 the natural morphisms we have an embedding ϕ : R → lim ← − R n such that ϕ(I) is an ideal. Moreover, M(I) = lim ← − M(R n ).
Proof. If x ∈ J n then it annihilates I n for every n, whence x ∈ I ⊥ . But then x = 0 since I is essential and R is semi-prime. Thus our assumptions imply that J n = 0.
The quotient morphisms ϕ n : R → R n satisfy π n • ϕ n = ϕ n−1 for all n, and therefore define a morphism ϕ : R → lim ← − R n such that ρ n • ϕ = ϕ n for all n, where ρ n : lim ← − R n → R n are the coordinate evaluations. Since ker ϕ = J n = 0, this is an embedding.
Observe that ϕ n (I m ) = (I m + J n )/J n is an ideal in R n for every n and m. Since ker π n = J n−1 /J n we see moreover that ϕ n (I m ) ∩ ker π n = 0 for n > m. Consequently (ϕ n (I m )) is a coherent sequence of ideals in (R n ), all isomorphic for n ≥ m, thus giving rise to the m−constant ideal ϕ(I m ) in lim ← − R n . It follows from this that ϕ(I) is an ideal in lim ← − R n , isomorphic to I (and contained in the quasi-constant ideal of lim ← − R n ). Since by assumption ρ n (ϕ(I)) = ϕ n (I) = (I + J n )/J n = R n , it follows from Theorem 6.1 that M(I) = lim ← − M(R n ).
Corollary 6.7. Let (I n ) and (J n ) be two sequences of ideals in a semi-prime σ−unital ring R, one increasing, the other decreasing, but such that I n ∩ J n = 0 for every n. If I n = R and each quotient R n = R/J n is unital, then with π n : R n → R n−1 the natural morphisms we have an embedding of R as an ideal in lim ← − R n such that M(R) = lim ← − R n . Examples 6.8. Let (e n ) be a countable approximate unit consisting of central elements in a semi-prime ring R. Assuming, as we may, that e n e n+1 = e n for all n we consider the principal ideals I n = e n R and J n = (1 − e n+1 )R in R. Note that e n ∈ I n , whereas J n consists of elements of the form x − e n+1 x, where x ∈ R. Evidently I n ∩ J n = 0 and I n = R. Moreover R n = R/J n is unital with unit e n+1 + J n . It follows from Corollary 6.7 that M(R) = lim ← − R n . Along the same lines we may assume that the semi-prime ring R has a countable approximate unit (e n ), such that e n+1 x = x = xe n+1 for every x in the principal ideal I n generated by e n . This could be accomplished by placing suitable finiteness conditions on I n , for example finite dimensionality. As before we let J n be the principal ideal in R generated by 1 − e n+1 (consisting of sums of elements of the form x − xe n+1 , y − e n+1 y and xy − xe n+1 y, with x, y in R). By assumption I n ∩ J n = 0 and I n = R. Moreover, each quotient R n = R/J n is unital with unit e n+1 + J n , so again M(R) = lim ← − R n . Note that in both examples the quotients R n−1 = R/J n−1 can be identified with the local rings R en (= e n Re n as an additive group, but with the product (e n xe n )(e n ye n ) = e n xe n ye n ). We know from [22] that every extension is determined (up to an obvious isomorphism) by

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