On a question of Faith in commutative endomorphism rings

Clark, John

doi:10.1090/S0002-9939-1986-0854017-6

On a question of Faith in commutative endomorphism rings
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Proc. Amer. Math. Soc. 98 (1986), 196-198 Request permission

Abstract:

Given a commutative ring $R$, let $Q(R)$ denote its maximal ring of quotients and, for any ideal $I$ of $R$, let $\operatorname {End} (I)$ denote the ring of $R$-endomorphisms of $I$. It is known that if $Q(R)$ is a self-injective ring then $\operatorname {End} (I)$ is commutative for each ideal $I$ of $R$. Carl Faith has asked if the converse holds. It does if $R$ is either Noetherian or has no nontrivial nilpotent elements but here we produce an example to show that it does not hold in general.

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