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Invariant ideals of abelian group algebras under the multiplicative action of a field. II


Authors: J. M. Osterburg, D. S. Passman and A. E. Zalesskii
Journal: Proc. Amer. Math. Soc. 130 (2002), 951-957
MSC (2000): Primary 16S34, 12E20
DOI: https://doi.org/10.1090/S0002-9939-01-06338-9
Published electronically: November 9, 2001
MathSciNet review: 1873766
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Abstract: Let $D$ be a division ring and let $V=D^n$ be a finite-dimensional right $D$-vector space, viewed multiplicatively. If $G=D^\bullet$ is the multiplicative group of $D$, then $G$ acts on $V$ and hence on any group algebra $K[V]$. In this paper, we completely describe the semiprime $G$-stable ideals of $K[V]$, and conclude that these ideals satisfy the ascending chain condition. As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields (handled in Part I).


References [Enhancements On Off] (What's this?)

  • [BE] C. J. B. Brookes and D. M. Evans, Augmentation modules for affine groups, Math. Proc. Cambridge Philos. Soc. 130 (2001), 287-294. CMP 2001:06
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Additional Information

J. M. Osterburg
Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221
Email: James.Osterburg@math.uc.edu

D. S. Passman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: Passman@math.wisc.edu

A. E. Zalesskii
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
Email: A.Zalesskii@uea.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-01-06338-9
Received by editor(s): October 3, 2000
Published electronically: November 9, 2001
Additional Notes: The first author’s research was supported by the Taft Committee of the University of Cincinnati. The second author’s research was supported in part by NSF Grant DMS-9820271. Much of this work was performed during the third author’s visit to the University of Wisconsin-Madison, made possible by the financial support of EPSRC
Communicated by: Lance W. Small
Article copyright: © Copyright 2001 American Mathematical Society