Question concerning finite rings

Question

Let $R$ be a finite ring. Is it possible that $R$ has an element $a\in R$ such that $a$ is a left divisor of zero and $a$ is not right divisor of zero?

Thanks.

Answer 1

Let $S$ be a finite semigroup of left zeroes ($ab=a$ for all $a,b\in S$), $F$ a finite field, $FS$ the semigroup ring of $S$ over $F$. Then $a(b-c)=0$, so every $a$ is a left divisor of zero but not a right divisor.

(Thanks @rschwieb for a remark)

Answer 2

Hints (which come under the assumption that the ring has identity):

On a finite set $X$, a mapping from $X$ to $X$ is 1-1 iff it is onto.

Suppose $a$ is not a right divisor of zero.

The mapping $r\mapsto ra$ is a homomorphism of left $R$ modules from $R$ to $R$. By an elementary proposition, all left module homomorphisms from $R$ to $R$ have this form.

What do the hypotheses say about this map? Might it have an inverse?