Let $A$ be an artinian ring. It is well known that for an element $x$ in $R$ the right annihilator $Ann\_r(x)$ is non trivial (i.e. contains a nonzero element ) if and only if the left annihilator $Ann\_l(x)$ is non trivial. 


If we assume that the ring $A$ is finite is true that the cardinality of $Ann\_r(x)$ equals 
the cardinality of $Ann\_l(x)$?


Note: If the ring $A$ is finite and semisimple, them it is a direct sum of full matrix rings over fields and in this case it is true. I think it may help to think first in full matrix rings over finite commutative rings. 



Annihilators in  artinian ring
No. Let $k$ be a finite field. Let $A$ be the non-commutative $k$ algebra generated by $x$ and $y$ subject to $x^2=0$, $y^2=0$, $yx=0$. So $\dim\_k A = 4$, with basis $1$, $x$, $y$, $xy$. Look at the element $y$: Its left annihilator has basis $(y, xy)$ and its right annihilator has basis $(x, y, xy)$. 


