Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-26T22:08:20.414Z Has data issue: false hasContentIssue false
  • English
  • Français

A Decomposition of Rings Generated by Faithful Cyclic Modules

Published online by Cambridge University Press:  20 November 2018

Gary F. Birkenmeier
Gary F. Birkenmeier*
Affiliation:
Department of Mathematics University of Southwestern Louisiana Lafayette, Louisiana 70504
Rights & Permissions [Opens in a new window]

Abstract

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 said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 3 , 01 September 1989 , pp. 333 - 339
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Birkenmeier, G. F., Self-injective rings and the minimal direct summand containing the nilpotents, Comm. Algebra 4 (1976), 705–721.Google Scholar
2. Birkenmeier, G. F., Baer rings and quasi-continuous rings have a MDSN, Pac. J. Math. 97 (1981), 283–292.Google Scholar
3. Birkenmeier, G. F., A generalization of FPF rings, Comm. Alg. 17 (1989), 855–884.Google Scholar
4. Birkenmeier, G. F., Extensions of strongly right bounded rings, .Google Scholar
5. Birkenmeier, G. F. and R. P. Tucci, Homomorphic images and the singular ideal of a strongly right bounded ring, Comm. Alg. 16 (1988), 1099–1112.Google Scholar
6. Birkenmeier, G. F. and H. E. Heatherly, Embeddings of strongly right bounded rings and algebras, Comm. Alg. 17 (1989), 573–586.Google Scholar
7. Chatters, A. W. and C. R. Hajarnavis, Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford, 28 (1977), 61–80.Google Scholar
8. Chatters, A. W. and S. M. Khuri, Endomorphism rings of modules over non-singular CS rings, J. London Math. Soc. 21 (1980), 434–444.Google Scholar
9. Clark, W. E., Twisted matrix units semigroup algebras, Duke Math. J. 34 (1967), 417–423.Google Scholar
10. Faith, C., Injective quotient rings of commutative rings, Springer Lecture Notes 700, Springer-Verlag, Berlin, 1979.Google Scholar
11. Faith, C. and S. Page, “FPF Ring Theory: Faithful Modules and Generators of Mod-R” London Mathematical Society Lecture Note Series 88, Cambridge University Press, Cambridge, 1984.Google Scholar
12. Faticoni, T. G., FPFrings /: The noetherian case, Comm. Algebra 13 (1985), 2119–2136.Google Scholar
13. Goodearl, K. R., “Ring Theory: Nonsingular Rings and Modules,” Marcel Dekker, New York, 1976. , “Von Neumann Regular Rings,” Pitman, London, 1979.Google Scholar
14. Hajarnavis, C. R. and N. C. Norton, On dual rings and their modules, J. Algebra 93 (1985), 253–266.Google Scholar
15. Kaplansky, I., “Rings of Operators,” W. A. Benjamin, New York, 1968.Google Scholar
16. Page, S. S., FPF rings and some conjectures of C. Faith, Canad. Math. Bull. 26 (1983), 257–259.Google Scholar
17. Stenstrom, B., “Rings of Quotients,” Springer-Verlag, New York, 1975.Google Scholar