Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T00:52:42.274Z Has data issue: false hasContentIssue false
  • English
  • Français

Endomorphism Rings and Gabriel Topologies

Published online by Cambridge University Press:  20 November 2018

Soumaya Makdissi Khuri
Soumaya Makdissi Khuri*
Affiliation:
American University of Beirut, Beirut, Lebanon
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 basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR(M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod ([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(RM) = d(BB), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 2 , 01 April 1984 , pp. 193 - 205
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Chatters, A. W. and Hajarnavis, C. R., Rings in which every complement right ideal is a direct summand, Quart. J. Math., Oxford (2) 28 (1977), 61–80.Google Scholar
2. Cozzens, J. H., Maximal orders and reflexive modules, Trans. Amer. Math. Soc. 219 (1976), 323–336.Google Scholar
3. Cozzens, J. H. and Faith, C., Simple Noetherian rings (Cambridge University Press, Cambridge-London-New York-Melbourne, 1975).CrossRefGoogle Scholar
4. Faith, C., Algebra: rings, modules and categories I, Grundlehren der Math. Wiss. 190 (Springer-Verlag, Berlin-Heidelberg-New York, 1973).Google Scholar
5. Faith, C., Injective quotient rings of commutative rings, Springer Lecture Notes in Mathematics 700, “Module Theory”, Proceedings, Seattle, (1977).Google Scholar
6. Jategaonkar, A. V., Endomorphism rings of torsionless modules, Trans. Amer. Math. Soc. 161 (1971), 457–466.Google Scholar
7. Khuri, S. M., Endomorphism rings and lattice isomorphisms, J. of Algebra 56 (1979), 401–408.Google Scholar
8. Leu, H. M., The ring of quotients of a module endomorphism ring, Tamkang J. of Math. 7 (1976), 77–86.Google Scholar
9. Müller, B. J., The quotient category of a Morita context, J. Algebra 28 (1974), 389–407.Google Scholar
10. Stenström, B., Rings of quotients, Grundlehren der Math. Wiss. 217 (Springer-Verlag, Berlin-Heidelberg-New York, 1975).Google Scholar
11. Zelmanowitz, J. M., Semisimple rings of quotients, Bull. Austral. Math. Soc. 19 (1978), 97–115.Google Scholar