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Proceedings of the American Mathematical Society

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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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Group algebras whose units satisfy a group identity. II
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by D. S. Passman PDF
Proc. Amer. Math. Soc. 125 (1997), 657-662 Request permission

Abstract:

Let $K[G]$ be the group algebra of a torsion group $G$ over an infinite field $K$, and let $U=U(G)$ denote its group of units. A recent paper of A. Giambruno, S. K. Sehgal, and A. Valenti proved that if $U$ satisfies a group identity, then $K[G]$ satisfies a polynomial identity, thereby confirming a conjecture of Brian Hartley. Here we add a footnote to their result by showing that the commutator subgroup $G’$ of $G$ must have bounded period. Indeed, this additional fact enables us to obtain necessary and sufficient conditions for $U(G)$ to satisfy an identity.
References
  • A. Giambruno, E. Jespers, and A. Valenti, Group identities on units of rings, Arch. Math. (Basel) 63 (1994), no. 4, 291–296. MR 1290601, DOI 10.1007/BF01189563
  • A. Giambruno, S. K. Sehgal and A. Valenti, Group algebras whose units satisfy a group identity, Proc. AMS, this issue.
  • 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
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Additional Information
  • D. S. Passman
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
  • MR Author ID: 136635
  • Email: passman@math.wisc.edu
  • Received by editor(s): August 31, 1995
  • Additional Notes: This research was supported by NSF Grant DMS-9224662.
  • Communicated by: Ronald M. Solomon
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 657-662
  • MSC (1991): Primary 16S34
  • DOI: https://doi.org/10.1090/S0002-9939-97-04024-0
  • MathSciNet review: 1415361