On perfect simple-injective rings

Authors:
W. K. Nicholson and M. F. Yousif

Journal:
Proc. Amer. Math. Soc. **125** (1997), 979-985

MSC (1991):
Primary 16D50, 16L30

DOI:
https://doi.org/10.1090/S0002-9939-97-03678-2

MathSciNet review:
1363179

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Abstract | References | Similar Articles | Additional Information

Abstract: Harada calls a ring right simple-injective if every -homomorphism with simple image from a right ideal of to is given by left multiplication by an element of . 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 is left perfect and right simple-injective, then is quasi-Frobenius if and only if the second socle of is countably generated as a left -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.

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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:
https://doi.org/10.1090/S0002-9939-97-03678-2

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

Article copyright:
© Copyright 1997
American Mathematical Society