Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

On perfect simple-injective rings

Author(s): W. K. Nicholson; M. F. Yousif
Journal: Proc. Amer. Math. Soc. 125 (1997), 979-985.
MSC (1991): Primary 16D50, 16L30
MathSciNet review: 1363179
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: Harada calls a ring $R$ right simple-injective if every $R$ -homomorphism with simple image from a right ideal of $R$ to $R$ is given by left multiplication by an element of $R$ . In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if $R$ is left perfect and right simple-injective, then $R$ is quasi-Frobenius if and only if the second socle of $R$ is countably generated as a left $R$ -module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings.

References:

1.: P. Ara, and J. K. Park, On continuous semiprimary rings, Comm. Alg. 19 (1991), 1945-1957. MR 92g:16006
2.: E. P. Armendariz and J. K. Park, , Arch. Math. 58 (1992), 24-33. MR 92m:16002
3.: Y. Baba and K. Oshiro, On a Theorem of Fuller, J. Algebra, 154 (1993), 86-94. MR 94b:16025
4.: J. A. Beachy, On quasi-artinian rings, J. London Math. Soc. (2) 3 (1971), 449-452. MR 44:244
5.: J.-E. Björk, Rings satisfying certain chain conditions, J. Reine Angew. Math. 245 (1970), 63-73. MR 43:3295
6.: N. Bourbaki, ``Algebra, Part I'', Hermann/Addison Wesley, 1974, p. 400. MR 50:6689
7.: V. Camillo, Commutative rings whose principal ideals are annihilators, Portugaliae Math. 46 (1) (1989), 33-37. MR 90e:13003
8.: J. Clark and D. V. Huynh, A note on perfect selfinjective rings, Quart. J. Math. Oxford (2) 45 (1994), 13-17. MR 95a:16005
9.: C. Faith, When self-injective rings are QF: A report on a problem, Centre Recerca Matemàtica Institut d'Estudis Catalans (Spain), 1990.
10.: K. R. Fuller, On indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115-135. MR 40:186
11.: C. R. Hajarnavis and N. C. Norton, On dual rings and their modules, J. Algebra 93 (1985), 253-266. MR 86i:16016
12.: M. Harada, On almost relative injective of finite length, preprint.
13.: L. S. Levy, Commutative rings whose homomorphic images are self-injective, Pacific J. Math. 18 (1966), 149-153. MR 33:2663
14.: W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 (1995), 77-93. MR 96i:16005
15.: B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373-389. MR 34:4305; MR 36:6443

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 16D50, 16L30

Retrieve articles in all Journals with MSC (1991): 16D50, 16L30

Additional Information:

W. K. Nicholson
Affiliation: Department of Mathematics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: wknichol@acs.ucalgary.ca

M. F. Yousif
Affiliation: Department of Mathematics, Ohio State University, Lima, Ohio 45804
Email: yousif.1@osu.edu

DOI: 10.1090/S0002-9939-97-03678-2
PII: S 0002-9939(97)03678-2
Keywords: Perfect ring, self-injective ring, quasi-Frobenius ring
Received by editor(s): April 24, 1995
Received by editor(s) in revised form: October 11, 1995
Additional Notes: The research of both authors was supported by NSERC Grant 8075 and by the Ohio State University.
Dedicated: Dedicated to Professor K. Varadarajan on the occasion of his sixtieth birthday
Communicated by: Ken Goodearl
Copyright of article: Copyright 1997, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|