Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-04-30T14:28:04.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Semilocal semigroup rings

Published online by Cambridge University Press:  18 May 2009

Jan Okniński
Jan Okniński
Affiliation:
University of Warsaw, 00-901 Warsaw, Poland.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Semilocal and related classes of group rings have been investigated by many authors (cf. [10]). In particular, the following results have been obtained.

Theorem A[4,10]. Let K be a field and G a group.

(i) If ch K = 0, then K[G] is semilocal if and only if G is finite.

(ii) If ch K = p>0 and G is locally finite, then K[G] is semilocal if and only if G contains a p-subgroup of finite index.

In the case of semigroup rings some stronger conditions have been studied. Munn examined the semisimple artinian situation [6]. Zelmanov showed that if K[G]is artinian then G must be finite [11].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 1 , January 1984 , pp. 37 - 44
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups (American Mathematical Society, 1961).Google Scholar
2.Faith, C., Algebra II: Ring Theory (Springer-Verlag 1976).CrossRefGoogle Scholar
3.Krempa, J. and Okniński, J., Semilocal, semiperfect and perfect tensor products, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 249–256.Google Scholar
4.Lawrence, J. and Woods, S. M., Semilocal group rings in characteristic zero, Proc. Amer. Math. Soc. 60 (1976), 8–10.CrossRefGoogle Scholar
5.Lee, Sin-Min, A condition for a semigroup ring to be local, Nanta Math. 11 (1978), 136–138.Google Scholar
6.Munn, W. D., On semigroup algebras, Proc. Cambridge Philos. Soc. 51 (1955), 1–15.CrossRefGoogle Scholar
7.Okniński, J., On spectrally finite algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 515–519.Google Scholar
8.Okniński, J., Spectrally finite and semilocal group rings, Comm. Algebra 8 (1980), 533–541.CrossRefGoogle Scholar
9.Okniński, J., Artinian semigroup rings, Comm. Algebra 10 (1982), 109–114.CrossRefGoogle Scholar
10.Passman, D. S., The algebraic structure of group rings. (Interscience, 1977).Google Scholar
11.Zelmanov, E. I., Semigroup algebras with identities, Siberian Math. J. 18 (1977), 787–798.Google Scholar