Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-07T01:58:59.435Z Has data issue: false hasContentIssue false
  • English
  • Français

Completely semisimple ring semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

John Hannah ,
J. S. Richardson  and
John Zeleznikow
John Hannah
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Australia, 3052
J. S. Richardson
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Australia3052
John Zeleznikow
Affiliation:
Department of Mathematical Sciences, Northern Illinois University at De Kalb, Illinois 60115, USA
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.

Let R be a ring in which the multiplicative semigroup is completely semisimple. If R has the maximum (respectively, minimum) condition on principal multiplicative ideals. then R is semiprime artinian (respectively, a direct sum of dense rings of finite-rank linear transformations of vector spaces over division rings).

MSC classification

Secondary: 20M25: Semigroup rings, multiplicative semigroups of rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 2 , December 1980 , pp. 150 - 156
Copyright
Copyright © Australian Mathematical Society 1980

References

Clifford, A. H. and Preston, G. B. (1961 and 1967), The algebraic theory of semigroups, Vols. I and II (American Mathematical Society, Providence, Rhode Island).Google Scholar
Goodearl, K. R. and Handelman, D. (1975), ‘Simple self-injective rings’, Comm. Algebra 3 (9), 797 834.CrossRefGoogle Scholar
Hall, T. E. (1970), ‘On the natural ordering of ƒ-classes and of idempotents in a regular semigroup’, Glasyow Math. J. II, 167–168.CrossRefGoogle Scholar
Howie, J. M. (1976), An introduction to semigroup theory (London Math. Soc. Monographs 7, Academic Press, London).Google Scholar
Jacobson, Nathan (1956), Structure of rings (American Mathematical Society, Providence, Rhode Island).CrossRefGoogle Scholar
McCoy, N. H. (1964), The theory of rings (Macmillan, New York).Google Scholar
Munn, W. D. (1957), ‘Semigroups satisfying minimal conditions’, Proc. Glasgow. Math. Assoc. 3, 145–152.CrossRefGoogle Scholar
Petrich, Mario (1974), Rings and semigroups (Lecture Notes in Mathematics 380, Springer-Verlag, Berlin, Heidelberg, New York).CrossRefGoogle Scholar
von Neumann, J. (1936), ‘On regular rings’, Proc. Nat. Acad. Sci. U.S.A. 22. 707 713.CrossRefGoogle ScholarPubMed