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The Jacobson radical of a band ring

Published online by Cambridge University Press:  24 October 2008

W. D. Munn
W. D. Munn
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
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Extract

A band is a semigroup in which every element is idempotent. In this note we give an explicit description of the Jacobson radical of the semigroup ring of a band over a ring with unity. It is shown, further, that this radical is nil if and only if the Jacobson radical of the coefficient ring is nil. For the particular case of a normal band (see below for the definition) the Jacobson radical of the band ring is nilpotent if and only if the Jacobson radical of the coefficient ring is nilpotent; but this result does not extend to arbitrary bands.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 2 , March 1989 , pp. 277 - 283
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Green, J. A. and Rees, P.. On semigroups in which xr = x. Proc. Cambridge Philos. Soc. 48 (1952), 35–40.CrossRefGoogle Scholar
[2]Howin, J. M.. An Introduction to Semigroup Theory (Academic Press, 1976).Google Scholar
[3]Kelarev, A. V. and Volkov, M. V.. On semigroup graded rings and their radicals. (Research announcement, Colloquium on Semigroups, Szeged, 1987.)Google Scholar
[4]McAlister, D. B.. Rings related to completely 0-simple semigroups. J. Austral. Math. Soc. 12 (1971), 257–274.CrossRefGoogle Scholar
[5]McAlister, D. B.. Representations of semigroups by linear transformations II. Sernigroup Forum 2 (1971), 283–320.CrossRefGoogle Scholar
[6]Teissier, M.. Sur l'algèbre d'un demi-groupe fini simple II. Cas général. C.R. Acad. Sci. Paris 234 (1952), 2511–2513.Google Scholar
[7]Teply, M. L., Turman, E. G. and Quesada, A.. On semisimple semigroup rings. Proc. Amer. Math. Soc. 79 (1980), 157–163.CrossRefGoogle Scholar