Peikert's framework for attacks on R-LWE: What "reduction modulo q" means? I am reading Peikert's paper [Pei16] about secure instantiating of R-LWE problem. In section 3.1, The author gives a new attack framework by using "reduction modulo an ideal divisor $\mathfrak{q}$ of the modulus $qR$". 
I'm a little new to Algebraic number theory and so I did not understand this framework especially the following part:

"Let $\mathfrak{q} ⊆ R$ be an ideal divisor of $qR$, having norm $N(\mathfrak{q}) := |R/\mathfrak{q}|$, and let $ψ$ be a continuous error
distribution over $K_\mathbb{R}$. Given Ring-LWE samples
$(a_i, b_i = s · a_i + e_i) \in R_q × K_\mathbb{R}/qR$
where $a_i ← R_q$ and $e_i ← ψ$, we can reduce them modulo $\mathfrak{q}$ to obtain samples $(a'_i = a_i \mod \mathfrak{q} , b'_i = b_i \mod \mathfrak{q}) \in R/{\mathfrak{q}} × K_\mathbb{R}/\mathfrak{q}$"

What "reduction modulo $\mathfrak{q}$" means? Can anyone help clarify things for me? Giving an example would be perfect.
Recall that any ideal $I$ of $R$ is in particular an additive subgroup of $R$, and that the quotient $R/I$ is the collection of cosets $a + I = \{a + i : i \in I\}$ for each $a \in R$.  Two cosets $a+I, b+I$ are equal iff $a-b \in I$.  The map that sends $a \in R$ to $a + I \in R/I$ is called "reduction modulo $I$," and often written "$a \bmod I$."  The quotient $R/I$ is a ring via the induced ring operations from $R$ itself: $(a+I) + (b+I) = (a+b) + I$ and similarly for multiplication.

Now suppose we have some ideal divisor $J$ of $I$, i.e., $J \supseteq I$.  The "natural" or "reduction mod $J$" homomorphism from $R/I$ to $R/J$ simply sends $a+I$ to $a+J$.  This is well defined because if $a+I = b+I$, then $a+J=b+J$ because $a - b \in I \subseteq J$.  (You can easily check that this is both an additive and multiplicative homomorphism.)

In the present setting, we have $I=qR$ and an ideal divisor $J = \mathfrak{q} \supseteq I$.  For a Ring-LWE sample $(a_i, b_i = s \cdot a_i + e_i)$, we should view $a_i \in R/qR$ as the coset $a_i + qR$.  Reducing it modulo $\mathfrak{q}$ just means sending it to the coset $a'_i = a_i + \mathfrak{q} \in R/\mathfrak{q}$.  Similarly, we send the coset $b_i + qR$ to the coset \begin{equation} b'_i = b_i + \mathfrak{q} = (s \cdot a_i + e_i) + \mathfrak{q} = (s \cdot a'_i + e_i) + \mathfrak{q}. \end{equation}
Note that the last equation holds because $s \in R$, $\mathfrak{q}$ is an ideal, and $(a_i - a'_i) = \mathfrak{q}$, so $s \cdot (a_i - a'_i) \subseteq \mathfrak{q}$.

The upshot of all this is that we can transform "mod $qR$" Ring-LWE samples $(a_i, b_i)$ into "mod $\mathfrak{q}$" samples $(a'_i, b'_i)$.  However, as the paper notes, if the error distribution $\psi$ is very "smooth" modulo $\mathfrak{q}$---i.e., if $e + \mathfrak{q}$ is very close to uniform for $e \gets \psi$---then the resulting sample is useless: $(a'_i, b'_i)$ is just uniform junk, and it's impossible (information theoretically) to recover anything about $s$ from it.