# Annihilators in matrix rings

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?

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No, this isn't true in general, but it's true in principal ideal rings. Consider $R=\mathbb{F}_2[x,y]/(x,y)^2$ (a finite ring that isn't a principal ideal ring, which I found in this answer by Zev). The matrix
$$A=\pmatrix{x&x\\y&y}$$
annihilates the $16$ vectors that have no constant term, both on the left and on the right, but on the right it also annihilates the $16$ vectors that have a constant term in both components, for a total of $32$. Thus the left annihilator in $M_n(R)$ has $256$ elements, whereas the right annihilator has $1024$.
However, if $R$ is an elementary divisor ring, we can bring $A$ into Smith normal form using invertible matrices. Since this is diagonal, its left and right annihilators are the same up to transposition, and the invertible matrices don't change their cardinality. (See this MO question.)