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.