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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
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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

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