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 is a nontrivial idempotent in *KG* then its trace is a rational number , 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 is invertible in *R*, then *RG* has no nontrivial idempotents.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0427361-9

Keywords:
Group rings,
idempotent,
trace

Article copyright:
© Copyright 1977
American Mathematical Society