## 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**
—, - 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

*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.

## 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