Self-injective ring

Remarks

Example: $R = \Z/n\Z$ for $n \ge 1$ is self-injective. By Baer's criterion it suffices to extend maps $f$ from an ideal $d\Z/n\Z$, with $d \mid n$, to $R$. The image $a = f(d)$ satisfies $(n/d) a = 0$, so $a = db$ for some $b \in R$, and $x \mapsto bx$ extends $f$. Non-example: $\Z$ is not self-injective. The map $2\Z \to \Z$, $2m \mapsto m$, has no extension $g \colon \Z \to \Z$, since $2 g(1) = 1$ has no solution in $\Z$. Left and right self-injectivity are different conditions in general. A ring is quasi-Frobenius if and only if it is left Noetherian and left self-injective.