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Rings Characterized by their Cyclic Modules

Published online by Cambridge University Press:  20 November 2018

P. F. Smith
P. F. Smith*
Affiliation:
University of Glasgow, Glasgow, Scotland
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A ring R (with identity element) is called a right PCI-ring if and only if every proper cyclic right R-module is injective; that is, if C is a cyclic right R-module then either C ≌ R or C is injective. Faith [3, Theorems 14 and 17] (or see [2, Proposition 6.12 and Theorem 6.17]) proved that if a ring R is a right PCI-ring then R is semiprime Artinian or R is a simple right semihereditary right Ore domain. These latter rings we shall call simple rightPCI-domains. Examples of non-Artinian simple right PCI-domains were produced by Cozzens [1]. The object of this paper is to examine rings with similar properties and thus extend Faith's results.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 1 , 01 February 1979 , pp. 93 - 111
Copyright
Copyright © Canadian Mathematical Society 1979

References

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