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On a class of Noetherian algebras

Published online by Cambridge University Press:  14 November 2011

E. Jespers  and
J. Okniński
E. Jespers
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada, A1C 5S7
J. Okniński
Affiliation:
Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
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Extract

A class of Noetherian semigroup algebras K[S] is described. In particular, we show that, for any submonoid S of the semigroup Mn of all monomial n × n matrices over a polycyclic-by-finite group G, K[S] is right Noetherian if and only if S satisfies the ascending chain condition on right ideals. This is then used to prove that every prime homomorphic image of a semigroup algebra of a finitely generated Malcev nilpotent semigroup S satisfying the ascending chain condition on right ideals is left and right Noetherian.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1185 - 1196
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Clifford, A. H. and Preston, G. B.. The algebraic theory of semigroups, vol. I (Providence, Rf: AMS, 1961).Google Scholar
2Fountain, J. and Petrich, M.. Completely 0-simple semigroups of quotients. J. Algebra 101 (1986), 365–402.CrossRefGoogle Scholar
3Herstein, L N.. Rings with involutions (University of Chicago Press, 1976).Google Scholar
4Jespers, E. and Okniński, J.. Nilpotent semigroups and semigroup algebras. J. Algebra 169 (1994), 984–1011.CrossRefGoogle Scholar
5Jespers, E. and Okniński, J.. Noetherian semigroup algebras. J. Algebra 218 (1999), 543–562.CrossRefGoogle Scholar
6Jespers, E. and Okniński, J.. Submonoids of polycyclic-by-finite groups and their algebras. Algebras Representat. Theory. (In the press.)Google Scholar
7Krause, G. R. and Lenagan, T. H.. Growth of algebras and Gelfand-Kirillov dimension (Boston, MA: Pitman, 1985).Google Scholar
8McConnell, J. C. and Robson, J. C.. Noncommutative Noetherian rings (Wiley, 1987).Google Scholar
9Okniński, J.. Semigroup algebras (New York: Dekker, 1991).Google Scholar
10Okniński, J.. Semigroups of matrices (World Scientific, 1998).CrossRefGoogle Scholar
11Passman, D. S.. The algebraic structure of group rings (Wiley, 1977).Google Scholar
12Quinn, D.. Group-graded rings and duality. Trans. Am. Math. Soc. 292 (1985), 155–167.CrossRefGoogle Scholar
13Reutenauer, C.. An Ogden-like iteration lemma for rational languages. Ada Informatica 13 (1980), 189–197.CrossRefGoogle Scholar
14Sapir, M.. Combinatorics on words with applications. Preprint LITP, 95/32, 1995.Google Scholar