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Rings characterized by cyclic modules

  • Dinh van Huynh (a1) and Phan Dan (a1)
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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.

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References
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1.Chatters, A. W., A characterisation of right Noetherian rings, Quart. J. Math. Oxford Ser. (2) 33 (1982), 6569.
2.Cozzens, J. H. and Faith, C., Simple Noetherian rings (Cambridge University Press, 1975).
3.Damiano, R. F., A right PCI ring is right Noetherian, Proc. Amer. Math. Soc. 77 (1979), 1114.
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.
5.Faith, C., Algebra II: Ring theory (Springer, 1976).
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.
7.Michler, G. O. and Villiamayor, O. E., On rings whose simple modules are injective, J. Algebra 25 (1973), 185201.
8.Smith, P. F., Rings characterized by their cyclic modules, Canad. J. Math. 31 (1979), 93111.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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