Trivial extension of a ring with balanced condition
HTML articles powered by AMS MathViewer
- by Hideaki Sekiyama PDF
- Proc. Amer. Math. Soc. 77 (1979), 1-6 Request permission
Abstract:
A ring R is called QF-1 if every faithful R-module is balanced. In this paper we study commutative QF-1 rings. It is shown that a commutative QF-1 ring is local if and only if it is uniform. It is well known that commutative artinian QF-1 rings are QF, but Osofsky has constructed a nonartinian nonnoetherian commutative injective cogenerator, so QF-1, ring which is a trivial extension of a valuation ring. It is shown that if a trivial extension of a valuation ring is QF-1, then it has a nonzero socle. Furthermore such rings become injective cogenerator rings under certain conditions.References
- Victor P. Camillo, Balanced rings and a problem of Thrall, Trans. Amer. Math. Soc. 149 (1970), 143β153. MR 260794, DOI 10.1090/S0002-9947-1970-0260794-0 β, A property of QF-1 rings (preprint).
- S. E. Dickson and K. R. Fuller, Commutative $\textrm {QF}-1$ artinian rings are $\textrm {QF}$, Proc. Amer. Math. Soc. 24 (1970), 667β670. MR 252426, DOI 10.1090/S0002-9939-1970-0252426-8
- Carl Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR 0427349
- Carl Faith, Self-injective rings, Proc. Amer. Math. Soc. 77 (1979), no.Β 2, 157β164. MR 542077, DOI 10.1090/S0002-9939-1979-0542077-8
- Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. Homological algebra of trivial extensions of abelian categories with applications to ring theory. MR 0389981
- Masatosi Ikeda and Tadasi Nakayama, On some characteristic properties of quasi-Frobenius and regular rings, Proc. Amer. Math. Soc. 5 (1954), 15β19. MR 60489, DOI 10.1090/S0002-9939-1954-0060489-9
- Bruno J. MΓΌller, On Morita duality, Canadian J. Math. 21 (1969), 1338β1347. MR 255597, DOI 10.4153/CJM-1969-147-7
- B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373β387. MR 204463, DOI 10.1016/0021-8693(66)90028-7
- Claus Michael Ringel, Commutative $\textrm {QF}-1$ rings, Proc. Amer. Math. Soc. 42 (1974), 365β368. MR 344283, DOI 10.1090/S0002-9939-1974-0344283-X
- Hans Heiner Storrer, Epimorphismen von kommutativen Ringen, Comment. Math. Helv. 43 (1968), 378β401 (German). MR 242810, DOI 10.1007/BF02564404
- Hans H. Storrer, Epimorphic extensions of non-commutative rings, Comment. Math. Helv. 48 (1973), 72β86. MR 321977, DOI 10.1007/BF02566112
- Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $\textrm {QF}-3$ and $\textrm {QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR 0349740
- Hiroyuki Tachikawa, Commutative perfect $\textrm {QF}-1$ rings, Proc. Amer. Math. Soc. 68 (1978), no.Β 3, 261β264. MR 472903, DOI 10.1090/S0002-9939-1978-0472903-1
- R. M. Thrall, Some generalization of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173β183. MR 26048, DOI 10.1090/S0002-9947-1948-0026048-0
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 77 (1979), 1-6
- MSC: Primary 16A36
- DOI: https://doi.org/10.1090/S0002-9939-1979-0539618-3
- MathSciNet review: 539618