Orders in self-injective cogenerator rings
HTML articles powered by AMS MathViewer
- by Robert C. Shock PDF
- Proc. Amer. Math. Soc. 35 (1972), 393-398 Request permission
Abstract:
This note states necessary and sufficient conditions for a ring to be a right order in certain self-injective rings. A ring R is said to have the dense extension property if each R-homomorphism from a right ideal of R into R can be lifted to some dense right ideal of R. A right ideal K is rationally closed if for each $x \in R - K$ the set ${x^{ - 1}}K = \{ y \in R:xy \in K\}$ is not a dense right ideal of R. We state a major result. Let $\dim R$ denote the Goldie dimension of a ring R and $Z(R)$ the right singular ideal of R. Then R is a right order in a self-injective cogenerator ring if and only if R has the dense extension property, $Z(R)$ is rationally closed and the factor ring $R/Z(R)$ is semiprime with $\dim R/Z(R) = \dim R < \infty$.References
- Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI 10.1090/S0002-9947-1960-0157984-8
- Carl Faith, Lectures on injective modules and quotient rings, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR 0227206
- G. D. Findlay and J. Lambek, A generalized ring of quotients. I, II, Canad. Math. Bull. 1 (1958), 77–85, 155–167. MR 94370, DOI 10.4153/CMB-1958-009-3
- Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
- A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220. MR 111766, DOI 10.1112/plms/s3-10.1.201
- Oscar Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10–47. MR 245608, DOI 10.1016/0021-8693(69)90004-0
- J. P. Jans, On orders in quasi-Frobenius rings, J. Algebra 7 (1967), 35–43. MR 212050, DOI 10.1016/0021-8693(67)90066-X
- R. E. Johnson, The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895. MR 45695, DOI 10.1090/S0002-9939-1951-0045695-9
- Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Tadasi Nakayama, On Frobeniusean algebras. II, Ann. of Math. (2) 42 (1941), 1–21. MR 4237, DOI 10.2307/1968984
- A. C. Mewborn and C. N. Winton, Orders in self-injective semi-perfect rings, J. Algebra 13 (1969), 5–9. MR 245619, DOI 10.1016/0021-8693(69)90003-9
- B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373–387. MR 204463, DOI 10.1016/0021-8693(66)90028-7 R. C. Shock, Right orders in self-injective rings, Notices Amer. Math. Soc. 17 (1970), 561. Abstract #70T-A76.
- Robert C. Shock, Injectivity, annihilators and orders, J. Algebra 19 (1971), 96–103. MR 279135, DOI 10.1016/0021-8693(71)90117-7
- Hiroyuki Tachikawa, Localization and Artinian quotient rings, Math. Z. 119 (1971), 239–253. MR 308169, DOI 10.1007/BF01113398
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 393-398
- MSC: Primary 16A18
- DOI: https://doi.org/10.1090/S0002-9939-1972-0302683-6
- MathSciNet review: 0302683