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Proceedings of the American Mathematical Society

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

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Defeat of the $\textrm {FP}^ 2\textrm {F}$ conjecture: Huckaba’s example
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by Carl Faith PDF
Proc. Amer. Math. Soc. 116 (1992), 5-6 Request permission

Abstract:

A commutative ring $R$ is $F{P^2}F$ (resp. FPF) provided that all finitely presented (resp. finitely generated) faithful modules generate the category $\bmod - R$ of all $R$-modules. A conjecture of the author dating to the middle 1970s states that any $F{P^2}F$ ring $R$ has FP-injective classical quotient ring $Q = {Q_{cl}}(R)$. It was shown by the author (Injective quotient rings. II, Lecture Notes in Pure and Appl. Math., vol. 72, Dekker, New York, 1982, pp. 71-105) that FPF rings $R$ have injective $Q$ and by the author and P. Pillay (Classification of commutative FPF rings, Notas Math., vol. 4, Univ. de Murcia, Murcia, Spain, 1990) that $CF{P^2}F$ local rings (defined below) have FP-injective $Q$. The counterexample is a difficult example due to Huckaba of a strongly Prüfer ring without "Property A." (A ring with Property A was labelled a McCoy ring by the author.) This counterexample is $CF{P^2}F$ in the sense that every factor ring of $R$ is $F{P^2}F$.
References
  • 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
  • —, Associated primes, annihilator ideals, and Kasch-McCoy commutative rings (in memory of Robert Warfield), Comm. Algebra 119 (1991). —, Injective quotient rings. II, Lecture Notes in Pure and Applied Math., vol. 72, Dekker, New York, 1982, pp. 71-105.
  • Carl Faith and Poobhalan Pillay, Classification of commutative FPF rings, Notas de Matemática [Mathematical Notes], vol. 4, Universidad de Murcia, Secretariado de Publicaciones e Intercambio Científico, Murcia, 1990. MR 1091714
  • James A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, vol. 117, Marcel Dekker, Inc., New York, 1988. MR 938741
  • James A. Huckaba and James M. Keller, Annihilation of ideals in commutative rings, Pacific J. Math. 83 (1979), no. 2, 375–379. MR 557938, DOI 10.2140/pjm.1979.83.375
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 116 (1992), 5-6
  • MSC: Primary 13B30; Secondary 13C11
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1097343-3
  • MathSciNet review: 1097343