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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
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] Carl Faith, Injective modules and injective quotient rings, Lecture Notes in Pure and Applied Mathematics, vol. 72, Marcel Dekker, Inc., New York, 1982. MR 643796
  • [2] Carl Faith, Injective quotient rings of commutative rings, Module theory (Proc. Special Session, Amer. Math. Soc., Univ. Washington, Seattle, Wash., 1977) Lecture Notes in Math., vol. 700, Springer, Berlin, 1979, pp. 151–203. MR 550435
  • [3] Carl Faith, Self-injective rings, Proc. Amer. Math. Soc. 77 (1979), no. 2, 157–164. MR 542077, 10.1090/S0002-9939-1979-0542077-8
  • [4] Carl Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976. Ring theory; Grundlehren der Mathematischen Wissenschaften, No. 191. MR 0427349
  • [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] Carl Faith and Stanley Page, FPF ring theory, London Mathematical Society Lecture Note Series, vol. 88, Cambridge University Press, Cambridge, 1984. Faithful modules and generators of mod-𝑅. MR 754181

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Article copyright: © Copyright 1984 American Mathematical Society