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?



Annihilators in Matrix Rings 
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 $|k|^8$.


