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On the radical of semigroup algebras satisfying polynomial identities

Published online by Cambridge University Press:  24 October 2008

Jan Okniński
Affiliation:
Institute of Mathematics, University of Warsaw, 00–901 Warsaw, Poland

Extract

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebra K[S] of an arbitrary semigroup S over a field K in the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties of S or, more precisely, by some groups derived from S. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups. Math. Surveys of the Amer. Math. Soc. vol. 7 (Providence, 1961).Google Scholar
[2]Cohen, M. and Montgomery, S.. Group-graded rings, smash products, and group actions. Trans. Amer. Math. Soc. 282 (1984), 237258.Google Scholar
[3]Cohen, M. and Rowen, L. H.. Group graded rings. Comm. Algebra 11 (1983), 12531270.Google Scholar
[4]Domanov, O. I.. Identities of semigroup algebras of 0-simple semigroups. Siberian Math. J. 17 (1976), 14061407.Google Scholar
[5]Domanov, O. I.. The semisimplicity and identities of semigroup algebras of inverse semi-groups. Rings and Modules, Mat. Issled. Vyp. 38 (1976), 123131.Google Scholar
[6]Faith, C.. Algebra II, Ring Theory (Springer-Verlag, 1976).Google Scholar
[7]Herstein, I. N.. Noncommutative Rings. Carus Math. Monographs vol. 15 (Wiley, 1968).Google Scholar
[8]Jacobson, N.. Structure of Rings. Amer. Math. Soc. Coll. Publ. vol. 37 (Providence, 1968).Google Scholar
[9]Munn, W. D.. On commutative semigroup algebras. Math. Proc. Cambridge Philos. Soc. 93 (1983), 237246.Google Scholar
[10]Okniński, J.. Semilocal semigroup rings. Glasgow Math. J. 25 (1984), 3744.Google Scholar
[11]Okniński, J.. Strongly π-regular matrix semigroups. Proc. Amer. Math. Soc., 93 (1985), 215217.Google Scholar
[12]Parker, T. and Gilmer, R.. Nilpotent elements of commutative semigroup rings. Michigan Math. J. 22 (1975), 97108.Google Scholar
[13]Passman, D. S.. The Algebraic Structure of Group Rings (Wiley-Interscience, 1977).Google Scholar
[14]Schneider, H. and Weissglass, J.. Group rings, semigroup rings and their radicals. J. Algebra 5 (1967), 115.CrossRefGoogle Scholar
[15]Zelmanov, E. I.. Semigroup algebras with identities. Siberian Mat. J. 18 (1977), 787798.Google Scholar