Hostname: page-component-5d59c44645-lfgmx Total loading time: 0 Render date: 2024-02-22T09:11:32.352Z Has data issue: false hasContentIssue false

Rings characterized by cyclic modules

Published online by Cambridge University Press:  18 May 2009

Dinh van Huynh
Affiliation:
Institute of Mathematics, P.O. Box 631 Bo Hô, Hanoi, Vietnam
Phan Dan
Affiliation:
Institute of Mathematics, P.O. Box 631 Bo Hô, Hanoi, Vietnam
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A ring R is called right PCI if every proper cyclic right R-module is injective, i.e. if C is a cyclic right R-module then CRRR or CR is injective. By [2] and [3], if R is a non-artinian right PCI ring then R is a right hereditary right noetherian simple domain. Such a domain is called a right PCI domain. The existence of right PCI domains is guaranteed by an example given in [2]. As generalizations of right PCI rings, several classes of rings have been introduced and investigated, for example right CDPI rings, right CPOI rings (see [8], [6]). In Section 2 we define right PCS, right CPOS and right CPS rings and study the relationship between all these rings.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Chatters, A. W., A characterisation of right Noetherian rings, Quart. J. Math. Oxford Ser. (2) 33 (1982), 6569.Google Scholar
2.Cozzens, J. H. and Faith, C., Simple Noetherian rings (Cambridge University Press, 1975).Google Scholar
3.Damiano, R. F., A right PCI ring is right Noetherian, Proc. Amer. Math. Soc. 77 (1979), 1114.Google Scholar
4.Huynh, Dinh van, Dung, Nguyen V. and Smith, Patrick F., On rings characterized by their right ideals or cyclic modules, Proc. Edinburgh Math. Soc., to appear.Google Scholar
5.Faith, C., Algebra II: Ring theory (Springer, 1976).Google Scholar
6.Goel, S. C., Jain, S. K. and Singh, S., Rings whose cyclic modules are injective or projective, Proc. Amer. Math. Soc. 53 (1975), 1618.Google Scholar
7.Michler, G. O. and Villiamayor, O. E., On rings whose simple modules are injective, J. Algebra 25 (1973), 185201.Google Scholar
8.Smith, P. F., Rings characterized by their cyclic modules, Canad. J. Math. 31 (1979), 93111.Google Scholar