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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

1. Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79.Google Scholar
2. Cozzens, J. H. and Faith, C., Simple Noetherian rings (Cambridge University Press, 1975).Google Scholar
3. Faith, C., When are proper cyclics infective﹜ Pacific. J. Math. 45 (1973), 97–112.Google Scholar
4. Goldie, A. W., Semiprime rings with maximum condition, Proc. London Math. Soc. (3). 10 (1960), 201–220.Google Scholar
5. Goodearl, K. R., Singular torsion and the splitting properties, Amer. Math. Soc. Memoirs No. 124 (1972).Google Scholar
6. Michler, G. O. and Villamayor, O. E., On rings whose simple modules are injective, J. Algebr. 25 (1973), 185–201.Google Scholar
7. Osofsky, B. L., Non-infective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), 1383–1384.Google Scholar
8. Rosenberg, A. and Zelinsky, D., Finiteness of the injective hull, Math. Zeit. 70 (1959), 372–380.Google Scholar
9. Small, L. W., Semiheriditary rings, Bull. Amer. Math. Soc. 73 (1967), 656–658.Google Scholar