Annihilators in Matrix Rings  I think this is not a research question, but in stackExchange remained unanswered.
Let $R$  be a finite commutative ring. For $n>1$  consider the full matrix ring $M_n(R)$ . For a matrix $A\in M_n(R)$  is true that the cardinality of the left annihilator (in $M_n(R)$ ) of $A$  equals the cardinality of the right annhilator?
 A: Let $k$ be a finite field, and let $R := k[X,Y] / (X^2,XY,Y^2)$. Then $R = k \oplus kx \oplus ky$ is a finite ring of order $|k|^3$ with maximal ideal $\mathfrak{m} := kx \oplus ky$ of square zero.
Now let $A = \begin{pmatrix} 0 & x \newline 0 & y \end{pmatrix} \in M_2(R)$. Then
$\begin{pmatrix} 0 & x \newline 0 & y \end{pmatrix} \begin{pmatrix} a & b \newline c & d \end{pmatrix} = \begin{pmatrix} cx & dx \newline cy & dy\end{pmatrix}$ whereas $\begin{pmatrix} a & b \newline c & d \end{pmatrix}\begin{pmatrix} 0 & x \newline 0 & y \end{pmatrix}  = \begin{pmatrix} 0 & ax +by \newline 0 & cx + dy\end{pmatrix}$.
The equation $ax + by = 0$ with $a,b \in R$ implies $a, b \in \mathfrak{m}$ as can be seen by writing $a = \lambda + u, b = \mu + v$ for some $\lambda,\mu \in k$ and $u,v \in \mathfrak{m}$. 
Hence rann$(A) = \begin{pmatrix} R & R \newline \mathfrak{m} & \mathfrak{m}\end{pmatrix}$ has size $|k|^{10}$, whereas lann$(A) = \begin{pmatrix} \mathfrak{m} & \mathfrak{m} \newline \mathfrak{m} & \mathfrak{m}\end{pmatrix}$ has size $|k|^8$.
A: First note that $R$ as a finite ring is (like any Artinian ring) is a finite product of local rings: $R=\prod_i R_i$. Hence $M_n(R) =\prod_i M_n(R_i).$ So if we write $A=A_1 \times \cdots A_n$ with $A_i \in M_n(R_i)$, then $$\text{Ann}^l(A) = \text{Ann}^l(A_1) \times \cdots \text{Ann}^l(A_n)$$ and analogous for the right annulator. 

If each $R_i$ is a local principal ideal ring (or more generally an elementary divisor ring), then $$|\text{Ann}^l(A)|=|\text{Ann}^r(A)|.$$ 

Proof: By the above we may assume wlog $R=R_i$ is a local PIR. By a theorem of Kaplansky, $A$ has a Smith normal form, i.e. there are invertible $U,V$ and a diagonal matrix $D=\text{diag}(d_,...,d_n)$ such that $A=UDV$. Thus $XA=0$ is equivalent to $(XU)D=0$, showing 
$\text{Ann}^l(A)=\text{Ann}^l(D)U^{-1}$. In particular both sets have the same cardinality. Write $X=(x_1,...,x_n)$. Then $XD=(d_1x_1,...,d_nx_n)$. Hence 
$$\text{Ann}^l(D)=\lbrace (x_1,...,x_n) \mid \forall i: x_i \in \text{Ann}_R(d_i)^n \rbrace.$$
Analogously, $\text{Ann}^r(A)=V^{-1}\text{Ann}^r(D)$ and 
$$\text{Ann}^r(D)=\lbrace (y_1^T,...,y_n^T) \mid \forall i: y_i \in \text{Ann}_R(d_i)^n \rbrace.$$
Hence 
$$|\text{Ann}^l(A)| = \prod_{i=1}^n |\text{Ann}_R(d_i)|^n = |\text{Ann}^r(A)|$$
and the assertion follows. q.e.d. 
Remark: If $R$ is a (not necessarily finite) product of commutative principal ideal rings, then the proof above shows 

$\qquad\qquad \text{Ann}^r(A) = V^{-1}U^T \text{Ann}^l(A)^T$

