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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.

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up vote 3 down vote accepted

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 $(x, y, xy)$.

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thanks, is a very clever example. – Miguel May 29 '12 at 17:05

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