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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Commutative FPF rings arising as split-null extensions


Author: Carl Faith
Journal: Proc. Amer. Math. Soc. 90 (1984), 181-185
MSC: Primary 16A36; Secondary 16A14, 16A52
DOI: https://doi.org/10.1090/S0002-9939-1984-0727228-6
MathSciNet review: 727228
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Abstract: Let $ R = (B,E)$ be the split-null or trivial extension of a faithful module $ E$ over a commutative ring $ B$. $ R$ is an FPF ring iff the partial quotient ring $ B{S^{ - 1}}$ with respect to the set $ S$ of elements of $ B$ with zero annihilator in $ E$ is canonically the endomorphism ring of $ E$, that is $ B{S^{ - 1}} = {\operatorname{End}_B}E{S^{ - 1}}$, every finitely generated ideal with zero annihilator in $ E$ is invertible in $ B{S^{ - 1}}$, and $ E = E{S^{ - 1}}$ is an injective module over $ B$. The proof uses the author's characterization of commutative FPF rings [1] and also the characterization of self-injectivity of a split-null extension [3].


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DOI: https://doi.org/10.1090/S0002-9939-1984-0727228-6
Article copyright: © Copyright 1984 American Mathematical Society

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