Rings Characterized by their Cyclic Modules

Smith, P. F.

Canadian journal of mathematics, Tome 31 (1979) no. 1, pp. 93-111

Résumé

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.

Détail

DOI : 10.4153/CJM-1979-011-2

Bibliographie

[1] 1. Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79. Google Scholar

[2] 2. Cozzens, J. H. and Faith, C., Simple Noetherian rings (Cambridge University Press, 1975). Google Scholar

[3] 3. Faith, C., When are proper cyclics infective﹜ Pacific. J. Math. 45 (1973), 97–112. Google Scholar

[4] 4. Goldie, A. W., Semiprime rings with maximum condition, Proc. London Math. Soc. (3). 10 (1960), 201–220. Google Scholar

[5] 5. Goodearl, K. R., Singular torsion and the splitting properties, Amer. Math. Soc. Memoirs No. 124 (1972). Google Scholar

[6] 6. Michler, G. O. and Villamayor, O. E., On rings whose simple modules are injective, J. Algebr. 25 (1973), 185–201. Google Scholar

[7] 7. Osofsky, B. L., Non-infective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), 1383–1384. Google Scholar

[8] 8. Rosenberg, A. and Zelinsky, D., Finiteness of the injective hull, Math. Zeit. 70 (1959), 372–380. Google Scholar

[9] 9. Small, L. W., Semiheriditary rings, Bull. Amer. Math. Soc. 73 (1967), 656–658. Google Scholar