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Semi-Prime Rings Whose Homomorphic Images are Serial

Published online by Cambridge University Press:  20 November 2018

Lawrence S. Levy  and
Patrick F. Smith
Lawrence S. Levy
Affiliation:
University of Wisconsin, Madison, Wisconsin
Patrick F. Smith
Affiliation:
University of Glasgow, Glasgow, Scotland
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A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.

We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 691 - 695
Copyright
Copyright © Canadian Mathematical Society 1982

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