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

Passman, D.; Zalesskiĭ, A.

doi:10.1090/S0002-9939-01-06092-0

Invariant ideals of abelian group algebras under the multiplicative action of a field. I
HTML articles powered by AMS MathViewer

by D. S. Passman and A. E. Zalesskiĭ

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

DOI: https://doi.org/10.1090/S0002-9939-01-06092-0

Published electronically: November 9, 2001

PDF | Request permission

Abstract:

Let $D$ be a division ring and let $V=D^n$ be a finite-dimensional $D$-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 $K[V]$. Our goal is to completely describe the semiprime $G$-stable ideals of $K[V]$. 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.

References

C. J. B. Brookes and D. M. Evans, Augmentation modules for affine groups, Math. Proc. Cambridge Philos. Soc. 130 (2001), 287–294.
B. Hartley and A. E. Zalesskiĭ, Group rings of periodic linear groups, unpublished note (1995).
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 and representations of groups of Lie type, Trans. AMS 353 (2001), 2971–2982.
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, DOI 10.1007/978-94-011-0329-9_{9}