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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Invariant ideals of abelian group algebras under the multiplicative action of a field. II
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by J. M. Osterburg, D. S. Passman and A. E. Zalesskiĭ PDF
Proc. Amer. Math. Soc. 130 (2002), 951-957 Request permission

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
  • C. J. B. Brookes and D. M. Evans, Augmentation modules for affine groups, Math. Proc. Cambridge Philos. Soc. 130 (2001), 287–294.
  • Daniel R. Farkas and Robert L. Snider, Simple augmentation modules, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 177, 29–42. MR 1269288, DOI 10.1093/qmath/45.1.29
  • Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
  • D. S. Passman and A. E. Zalesskiĭ, Invariant ideals of abelian group algebras under the multiplicative action of a field, I, Proc. AMS, to appear.
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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
  • MR Author ID: 136635
  • Email: Passman@math.wisc.edu
  • A. E. Zalesskiĭ
  • Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
  • MR Author ID: 196858
  • Email: A.Zalesskii@uea.ac.uk
  • 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
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1873766