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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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  • [1] C. Faith, Injective quotient rings of commutative rings. II, Injective Modules and Injective Quotient Rings, Lecture Notes in Pure and Appl. Math., vol. 72, Dekker, New York, 1982. MR 643796 (83d:16023)
  • [2] -, Injective quotient rings of commutative rings. I, Module Theory, Lecture Notes in Math., vol. 700 (C. Faith and S. Wiegand, eds.), Springer-Verlag, Berlin and New York, 1979. MR 550435 (81a:13014)
  • [3] -, Self-injective rings, Proc. Amer. Math. Soc. 77 (1979), 157-164. MR 542077 (80i:16033)
  • [4] -, Algebra II: Ring theory, Springer-Verlag, Berlin and New York, 1976. MR 0427349 (55:383)
  • [5] -, Injective modules over Levitzki rings, Injective Modules and Injective Quotient Rings, Lecture Notes in Pure and Appl. Math., vol. 72, Dekker, New York, 1982.
  • [6] C. Faith and S. Page, FPF ring theory: faithful modules and generators of $ {\operatorname{Mod}} - R$, Lecture Notes of the London Math. Soc., Cambridge Univ. Press, 1984. MR 754181 (86f:16014)

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Additional Information

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

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