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Semirings with a completely 0-simple additive semigroup

Published online by Cambridge University Press:  09 April 2009

Mireille P. Grillet
Mireille P. Grillet
Affiliation:
4 Trianon Plaza, New OrleansLousiana 70125U.S.A.
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In a previous paper [2], we gave an explicit description of the structure of all semirings with a completely simple additive semigroup. The next step is then clearly to consider semirings with a completely 0-simple additive semigroup. We are able to classify these semirings according to the multiplicative nature of their additive zero. Let R be a semiring whose additive semigroup is completely 0-simple with zero ∞. First, if ∞ ∞ ≠ ∞, then the multiplication of R is trivial. Besides these trivial semirings, another class of semirings with a completely 0-simple additive semigroup can be easily obtained by adjoining an element ∞ which is together an additive zero and a multiplicative zero.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 4 , December 1974 , pp. 436 - 441
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Clifford, A. H.Preston, G. B., The algebraic theory of semigroups Vol. 1, Math. Surveys, No. 7 (Amer. Math. Soc., Providence, R. I., 1962), reprint 1964.Google Scholar
[2]Grillet, M. P., ‘Semirings with a completely simple additive semigroup’, (to appear).Google Scholar