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Rings with all modules residually finite

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Abstract

Define a ringA to be RRF (resp. LRF) if every right (resp. left) A-module is residually finite. Refer to A as an RF ring if it is simultaneously RRF and LRF. The present paper is devoted to the study of the structure of RRF (resp. LRF) rings. We show that all finite rings are RF. IfA is semiprimary, we show thatA is RRF ⇔A is finite ⇔A is LRF. We prove that being RRF (resp. LRF) is a Morita invariant property. All boolean rings are RF. There are other infinite strongly regular rings which are RF. IfA/J(A) is of bounded index andA does not contain any infinite family of orthogonal idempotents we prove:

  1. (i)

    A an RRF ring ⇔ A right perfect andA/J(A) finite (henceA/J(A) finite semisimple artinian).

  2. (ii)

    A an LRF ring ⇔ A left perfect andA/J(A) finite

IfA is one sided quasi-duo (left or right immaterial) not containing any infinite family of orthogonal idempotents then (i) and (ii) are valid with the further strengthening thatA/J(A) is a finite product of finite fields.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, T2N 1N4, Calgary, Alberta, Canada

    K. Varadarajan

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  1. K. Varadarajan
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Correspondence to K. Varadarajan.

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Varadarajan, K. Rings with all modules residually finite. Proc. Indian Acad. Sci. (Math. Sci.) 109, 345–351 (1999). https://doi.org/10.1007/BF02837992

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  • DOI: https://doi.org/10.1007/BF02837992

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