Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-30T18:56:08.129Z Has data issue: false hasContentIssue false
  • English
  • Français

On algebraic rings

Published online by Cambridge University Press:  17 April 2009

M. Chacron
M. Chacron
Affiliation:
University of Windsor, Windsor, Ontario, and University of British Columbia, Vancouver, British Columbia.
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 π-regular (periodic) if for each element x of R there is n = n(x) SO that xn = xn.a.xn (xn = xn.1.xn) (a depending on x). Let R be an algebraic algebra over a commutative ring F With identity. In this paper we prove that if every π-regular image of the ring F is periodic, then R is periodic. This result applies in particular to the algebraic rings R (over the integers) considered by Drazin and to the algebraic algebras R over algebraically prime fields. It extends a result of Drazin on torsin-free algebraic rings and a generalization by this author of Drazin's result.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 3 , December 1969 , pp. 385 - 389
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Chacron, M., “Certains anneaux périodiques”, Bull. Soc. Math. Belg. 20 (1968), 66–71.Google Scholar
[2]Chacron, M., “On quasi periodic rings”, J. Algebra 12 (1969), 49–60.CrossRefGoogle Scholar
[3]Chacron, M., “On a theorem of Herstein”, Canad. J. Math. (to appear).Google Scholar
[4]Chacron, M., “On a theorem of Procesi”, submitted to J. Algebra.Google Scholar
[5]Drazin, M.P., “Algebraic and diagonable rings”, Canad. J. Math. 8 (1956), 341–354.CrossRefGoogle Scholar
[6]Herstein, I.N., “A note on rings with central nilpotent elements”, Proc. Amer. Math. Soc. 5 (1954), 620.CrossRefGoogle Scholar
[7]Herstein, I.N., “A generalization of a theorem of Jacobson, III”, Amer. J. Math. 75 (1953), 105–111.CrossRefGoogle Scholar
[8]Herstein, I.N., “The structure of a certain class of rings”, Amer. J. Math. 75 (1953), 866–871.CrossRefGoogle Scholar