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Proceedings of the American Mathematical Society

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

 

 

On the trace of an idempotent in a group ring


Authors: Gerald H. Cliff and Sudarshan K. Sehgal
Journal: Proc. Amer. Math. Soc. 62 (1977), 11-14
MSC: Primary 16A26
DOI: https://doi.org/10.1090/S0002-9939-1977-0427361-9
MathSciNet review: 0427361
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Abstract: Let KG be the group ring of a polycyclic by finite group G over a field K of characteristic zero. It is proved that if $ e = \sum e(g)g$ is a nontrivial idempotent in KG then its trace $ e(1)$ is a rational number $ r/s,(r,s) = 1$, with the property that for every prime divisor p of s, G has an element of order p. This result is used to prove that if R is a commutative ring of characteristic zero, without nontrivial idempotents and G is a polycyclic by finite group such that no group order $ \ne 1$ is invertible in R, then RG has no nontrivial idempotents.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0427361-9
Keywords: Group rings, idempotent, trace
Article copyright: © Copyright 1977 American Mathematical Society