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

Authors:
D. S. Passman and A. E. Zalesskii

Journal:
Proc. Amer. Math. Soc. **130** (2002), 939-949

MSC (2000):
Primary 16S34, 12E20

Published electronically:
November 9, 2001

MathSciNet review:
1873765

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a division ring and let be a finite-dimensional -vector space, viewed multiplicatively. If is the multiplicative group of , then acts on and hence on any group algebra . Our goal is to completely describe the semiprime -stable ideals of . 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. Part I of this work is concerned with the latter situation, while Part II deals with arbitrary division rings.

**[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 CMP**2001:06****[HZ]**B. Hartley and A. E. Zalesski,*Group rings of periodic linear groups*, unpublished note (1995).**[P]**Donald S. Passman,*The algebraic structure of group rings*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR**470211****[PZ1]**D. S. Passman and A. E. Zalesski,*Invariant ideals of abelian group algebras and representations of groups of Lie type*, Trans. AMS**353**(2001), 2971-2982.**[Z]**A. E. Zalesskiĭ,*Group rings of simple locally finite groups*, Finite and locally finite groups (Istanbul, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, pp. 219–246. MR**1362812**, 10.1007/978-94-011-0329-9_9

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Additional Information

**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:
http://dx.doi.org/10.1090/S0002-9939-01-06092-0

Received by editor(s):
October 3, 2000

Published electronically:
November 9, 2001

Additional Notes:
The first author’s research was supported in part by NSF Grant DMS-9820271. Much of this work was performed during the second 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