Semiperfect FPF rings
HTML articles powered by AMS MathViewer
- by S. S. Page
- Proc. Amer. Math. Soc. 89 (1983), 395-401
- DOI: https://doi.org/10.1090/S0002-9939-1983-0715852-5
- PDF | Request permission
Abstract:
In this paper we derive some of the structure of semiperfect FPF rings. A ring is right FPF if every f.g. faithful right module is a generator. For semiperfect right and left FPF rings we show that if all one sided zero divisors are two sided zero divisors, then the classical and maximal quotient rings coincide (all four of them) and are self-injective. We show that if the intersection of the powers of the Jacobson radical is zero, then right and left regular elements are regular. Also, we show right FPF semiperfect rings contain the singular submodule of their injective hulls and that every finitely generated module contained in the injective hull and containing the ring is isomorphic to the ring.References
- A. W. Chatters and J. C. Robson, Decomposition of orders in semiprimary rings, Comm. Algebra 8 (1980), no. 6, 517–532. MR 561750, DOI 10.1080/00927878008822473
- 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 —, Injective quotient rings of commutative rings. II, Lecture Notes Pure Appl. Math., Vol. 72, Dekker, New York, 1982.
- Carl Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR 0427349
- N. Jacobson, Some remarks on one-sided inverses, Proc. Amer. Math. Soc. 1 (1950), 352–355. MR 36223, DOI 10.1090/S0002-9939-1950-0036223-1 J. Lambeck, Lectures on rings and modules, Blaisdell, New York, 1966; corrected reprint, Chelsea, New York, 1966, 1976.
- S. Page, Regular FPF rings, Pacific J. Math. 79 (1978), no. 1, 169–176. MR 526675
- S. S. Page, Semiprime and nonsingular FPF rings, Comm. Algebra 10 (1982), no. 20, 2253–2259. MR 676181, DOI 10.1080/00927878208822833 —, Semihereditary and full idempotent FPF rings (to appear).
- S. S. Page, FPF rings and some conjectures of C. Faith, Canad. Math. Bull. 26 (1983), no. 3, 257–259. MR 703392, DOI 10.4153/CMB-1983-040-8
- Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953
- Yuzo Utumi, On continuous rings and self injective rings, Trans. Amer. Math. Soc. 118 (1965), 158–173. MR 174592, DOI 10.1090/S0002-9947-1965-0174592-8
Bibliographic Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 395-401
- MSC: Primary 16A51; Secondary 16A08, 16A36, 16A48, 16A52
- DOI: https://doi.org/10.1090/S0002-9939-1983-0715852-5
- MathSciNet review: 715852