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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On a question of Faith in commutative endomorphism rings


Author: John Clark
Journal: Proc. Amer. Math. Soc. 98 (1986), 196-198
MSC: Primary 13B30; Secondary 13C11, 16A65
MathSciNet review: 854017
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0854017-6
PII: S 0002-9939(1986)0854017-6
Keywords: Endomorphism ring, maximal quotient ring, self-injective, trivial extension
Article copyright: © Copyright 1986 American Mathematical Society