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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
MathSciNet review: 1363179
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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.

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Additional Information

W. K. Nicholson
Affiliation: Department of Mathematics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

M. F. Yousif
Affiliation: Department of Mathematics, Ohio State University, Lima, Ohio 45804

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