Proceedings of the American Mathematical Society

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



Endomorphism rings of simple modules
over group rings

Author: Robert L. Snider
Journal: Proc. Amer. Math. Soc. 124 (1996), 1043-1049
MSC (1991): Primary 16S34, 20C05; Secondary 16K20, 16S50
MathSciNet review: 1327044
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Abstract | References | Similar Articles | Additional Information

Abstract: If $N$ is a finitely generated nilpotent group which is not abelian-by-finite, $k$ a field, and $D$ a finite dimensional separable division algebra over $k$, then there exists a simple module $M$ for the group ring $k[G]$ with endomorphism ring $D$. An example is given to show that this cannot be extended to polycyclic groups.

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

  • 1. P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, Cambridge, 1983. MR 696937
  • 2. Vasanti A. Jategaonkar, A multiplicative analog of the Weyl algebra, Comm. Algebra 12 (1984), no. 13-14, 1669–1688. MR 743310, 10.1080/00927878408823074
  • 3. D. L. Harper, Primitive irreducible representations of nilpotent groups, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 2, 241–247. MR 0447386
  • 4. D. L. Harper, Primitivity in representations of polycyclic groups, Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 1, 15–31. MR 569630, 10.1017/S0305004100057327
  • 5. J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
  • 6. Donald S. Passman, The algebraic structure of group rings, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1985. Reprint of the 1977 original. MR 798076
  • 7. Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR 674652
  • 8. J. E. Roseblade, Corrigenda: “Prime ideals in group rings of polycyclic groups” (Proc. London Math. Soc. (3) 36 (1978), no. 3, 385–447), Proc. London Math. Soc. (3) 38 (1979), no. 2, 216–218. MR 0491798

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Additional Information

Robert L. Snider
Affiliation: Department of Mathematics Virginia Tech Blacksburg, Virginia 24061-0123

Keywords: Group ring, endomorphism ring, division ring
Received by editor(s): October 17, 1994
Communicated by: Ken Goodearl
Article copyright: © Copyright 1996 American Mathematical Society