My question is:
Let $R$ be a ring with the unity $e$ and $a \in R$. If $a^{\circ}\triangleq\{x\in R \;| \;ax=0\}=\{0\}$, as the following conter-example given by Matthias Klupsch, we know that it is not left invertible in general. Then what additional conditions on $R$ or on $a$ will make that $a$ is left invertible?
If we have known $a$ is right invertilbe, then what will happen?
${\bf Notes:}$ Let $A$ and $B$ be two sets, and $f : A \to B$ be a mapping, then it is well-known that:
(1) $f$ is injective iff $f$ is left invertible.
(2) $f$ is surjective iff $f$ is right invertible.
Thanks!