Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T04:44:51.134Z Has data issue: false hasContentIssue false
  • English
  • Français

σ-Reflexive Semigroups and Rings

Published online by Cambridge University Press:  20 November 2018

M. Chacrono  and
G. Thierrin
M. Chacrono
Affiliation:
Carleton University, Ottawa, Ontario
G. Thierrin
Affiliation:
University of Western Ontario, London, Ontario
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.

We shall call a semigroup S a σ-reflexive semigroup if any subsemigroup H in S is reflexive (i.e. for all a, b ∈ S, ab ∈ H implies ba∊H ([2], [5]). It can be verified that any group G is a σ-reflexive semigroup if and only if any subgroup of G is normal. In this paper, we characterize subdirectly irreducible o-reflexive semigroups. We derive the following commutativity result: any generalized commutative ring R ([1]) in which the integers n=n(x, y) in the equation (xy)n = (yx)m can be taken equal to 1 for all x, y ∈ R must be a commutative ring.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 185 - 188
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Belluce, B., Jain, S. K., and Herstein, I. N.,Generalized commutative rings, Nagoya Math. J. 27 (1966), 1-5.Google Scholar
2. Clifford, A. H. and Preston, G. B., The Algebraic theory of semigroups, Math. Surveys No. 7, Vol. II, Amer. Math. Soc, 1967.Google Scholar
3. Hall, Marshall Jr, The theory of groups, MacMillan, New York, 1959.Google Scholar
4. Herstein, I. N., The structure of a certain class of rings, Amer. J. Math. 75 (1953), 866-871.Google Scholar
5. Thierrin, G., Contribution ? la théorie des anneaux et des demi-groupes, Comment. Math. Helv. 32(1957), 93-112.Google Scholar
6. Thierrin, G., Sur la structure des demi-groupes, Ann. Univ. d'alger, 3 (1956), 161-171.Google Scholar