The structure of Johns rings
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- by Carl Faith and Pere Menal
- Proc. Amer. Math. Soc. 120 (1994), 1071-1081
- DOI: https://doi.org/10.1090/S0002-9939-1994-1231294-8
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Erratum: Proc. Amer. Math. Soc. 125 (1997), 1247-1247.
Abstract:
In this paper we continue our study of right Johns rings, that is, right Noetherian rings in which every right ideal is an annihilator. Specifically we study strongly right Johns rings, or rings such that every $n \times n$ matrix ring ${R_n}$ is right Johns. The main theorem (Theorem 1.1) characterizes them as the left ${\text {FP}}$-injective right Noetherian rings, a result that shows that not all Johns rings are strong. (This first was observed by Rutter for Artinian Johns rings; see Theorem 1.2.) Another characterization is that all finitely generated right $R$-modules are Noetherian and torsionless, that is, embedded in a product of copies of $R$. A corollary to this is that a strongly right Johns ring $R$ is preserved by any group ring $RG$ of a finite group (Theorem 2.1). A strongly right Johns ring is right $FPF$ (Theorem 4.2).References
- Carl Faith, Algebra. I. Rings, modules, and categories, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 190, Springer-Verlag, Berlin-New York, 1981. Corrected reprint. MR 623254
- Carl Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR 0427349 —, Embedding modules in projectives, Advances in Non-Commutative Ring Theory, Lecture Notes in Math., vol. 951, Springer-Verlag, New York, 1982, pp. 21-39.
- Carl Faith, Embedding torsionless modules in projectives, Publ. Mat. 34 (1990), no. 2, 379–387. MR 1088885, DOI 10.5565/PUBLMAT_{3}4290_{1}6
- Carl Faith, Self-injective rings, Proc. Amer. Math. Soc. 77 (1979), no. 2, 157–164. MR 542077, DOI 10.1090/S0002-9939-1979-0542077-8
- Carl Faith, Commutative FPF rings arising as split-null extensions, Proc. Amer. Math. Soc. 90 (1984), no. 2, 181–185. MR 727228, DOI 10.1090/S0002-9939-1984-0727228-6 —, When self-injective rings are $QF$: A report on a problem, preprint.
- Carl Faith, Injective cogenerator rings and a theorem of Tachikawa, Proc. Amer. Math. Soc. 60 (1976), 25–30 (1977). MR 417237, DOI 10.1090/S0002-9939-1976-0417237-4
- Carl Faith, Semiperfect Prüfer rings and FPF rings, Israel J. Math. 26 (1977), no. 2, 166–177. MR 444693, DOI 10.1007/BF03007666
- Carl Faith, Rings with ascending condition on annihilators, Nagoya Math. J. 27 (1966), 179–191. MR 193107
- Carl Faith and Pere Menal, A counter example to a conjecture of Johns, Proc. Amer. Math. Soc. 116 (1992), no. 1, 21–26. MR 1100651, DOI 10.1090/S0002-9939-1992-1100651-0
- K. R. Goodearl, Embedding nonsingular modules in free modules, J. Pure Appl. Algebra 1 (1971), no. 3, 275–279. MR 299627, DOI 10.1016/0022-4049(71)90022-3
- Baxter Johns, Annihilator conditions in Noetherian rings, J. Algebra 49 (1977), no. 1, 222–224. MR 453808, DOI 10.1016/0021-8693(77)90282-4
- Saroj Jain, Flat and FP-injectivity, Proc. Amer. Math. Soc. 41 (1973), 437–442. MR 323828, DOI 10.1090/S0002-9939-1973-0323828-9
- Lawrence Levy, Torsion-free and divisible modules over non-integral-domains, Canadian J. Math. 15 (1963), 132–151. MR 142586, DOI 10.4153/CJM-1963-016-1
- E. A. Rutter, A characterization of $\textrm {QF}-3$ rings, Pacific J. Math. 51 (1974), 533–536. MR 366967
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 1071-1081
- MSC: Primary 16P40
- DOI: https://doi.org/10.1090/S0002-9939-1994-1231294-8
- MathSciNet review: 1231294