Commutativity of endomorphism rings of ideals. II

Abstract:

Let $R$ be a commutative ring. In (1), it was proved that a ring $R$ with noetherian total quotient ring is self-injective if and only if the endomorphism ring of every ideal is commutative. We prove here that if the ring is coherent and is its own total quotient ring, then $R$ is self-injective if and only if $\operatorname {Hom} (I,I) = R$ for every ideal $I$ of $R$.