Quasi-Frobenius ring

Remarks

The definition is left-right symmetric: a ring satisfying the two clauses is also right Noetherian and right self-injective, and it is left and right Artinian. Example: $\Z/n\Z$ for $n \ge 1$. It is finite, so clause 1 holds, and it is self-injective by Baer's criterion, so clause 2 holds. Non-example: $\Z$ is Noetherian but not self-injective, since the map $2\Z \to \Z$, $2m \mapsto m$, does not extend to $\Z$; clause 2 fails. Non-example: $k[x, y]/(x^2, xy, y^2)$ over a field $k$ is Artinian, hence Noetherian, but not self-injective, so clause 2 fails. Equivalent forms: $R$ is quasi-Frobenius if and only if every projective left $R$-module is injective, if and only if every injective left $R$-module is projective. Every Frobenius algebra is quasi-Frobenius.