Let $R$ be a commutative Noetherian ring. Then the proper quotients of $R$ are self-injective rings iff $R$ is one of the following types: 1) a Dedekind domain; 2) an Artinian principal ideal ring; 3) a local ring with maximal ideal $M$ such that $M^2=\{0\}$ and $R$ has composition length $2$.

- L. Levy. Commutative rings whose homomorphic images are self-injective. (1966) @ Main theorem