Algebraic elements in group rings
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- by I. B. S. Passi and D. S. Passman
- Proc. Amer. Math. Soc. 108 (1990), 871-877
- DOI: https://doi.org/10.1090/S0002-9939-1990-1007508-2
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Abstract:
In this brief note, we study algebraic elements in the complex group algebra ${\mathbf {C}}[G]$. Specifically, suppose $\xi \in {\mathbf {C}}[G]$ satisfies $f(\xi ) = 0$ for some nonzero polynomial $f(x) \in {\mathbf {C}}[x]$. Then we show that a certain fairly natural function of the coefficients of $\xi$ is bounded in terms of the complex roots of $f(x)$. For $G$ finite, this is a recent observation of [HLP]. Thus the main thrust here concerns infinite groups, where the inequality generalizes results of [K] and [W] on traces of idempotents.References
- Hyman Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976), 155–196. MR 432781, DOI 10.1007/BF01390137
- A. W. Hales, I. S. Luthar, and I. B. S. Passi, Partial augmentations and Jordan decomposition in group rings, Comm. Algebra 18 (1990), no. 7, 2327–2341. MR 1063137, DOI 10.1080/00927879008824023
- Irving Kaplansky, Fields and rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995. Reprint of the second (1972) edition. MR 1324341
- M. Susan Montgomery, Left and right inverses in group algebras, Bull. Amer. Math. Soc. 75 (1969), 539–540. MR 238967, DOI 10.1090/S0002-9904-1969-12234-2
- 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
- A. Weiss, Idempotents in group rings, J. Pure Appl. Algebra 16 (1980), no. 2, 207–213. MR 556161, DOI 10.1016/0022-4049(80)90017-1
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 871-877
- MSC: Primary 20C07; Secondary 16A27, 46L99
- DOI: https://doi.org/10.1090/S0002-9939-1990-1007508-2
- MathSciNet review: 1007508