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Orthodox semirings and rings

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

John Zeleznikow
John Zeleznikow
Affiliation:
Department of Mathematicx, Monash University, Clayton, Victoria 3168, Australia
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Abstract

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We show that in a regular ring (R, +, ·), with idempotent set E, the following conditions are equivalent: (i) (ii) (R, ·) is orthodox. (iii) (R, ·) is a semilattice of groups. These and other conditions are also considered for regular semigroups, and for semirings (S, +, · ), in which (S, +) is an inverse semigroup. Examples are given to show that they are not equivalent in these cases.

MSC classification

Secondary: 20M25: Semigroup rings, multiplicative semigroups of rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 1 , September 1980 , pp. 50 - 54
Copyright
Copyright © Australian Mathematical Society 1980

References

Chaptal, N. (1966), ‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Sér. A-B 262, 274–277.Google Scholar
Clifford, A. H. and Preston, G. B. (1961), The algebraic theory of semigroups, Vol. I (American Mathematical Society. Mathematical Surveys 7).Google Scholar
Grillet, M. O. P. (1970), ‘Subdivision rings of a semiring’, Fund Math. 67, 67–74.CrossRefGoogle Scholar
Karvellas, P. H. (1974), ‘Inversive semirings’, J. Austral. Math. Soc. 18, 277–287.Google Scholar
Kovacs, L. (1956), ‘A note on regular rings’, Publ. Math. Debrecen 4, 465–468.CrossRefGoogle Scholar