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A Problem of Herstein on Group Rings

Published online by Cambridge University Press:  20 November 2018

Edward Formanek
Edward Formanek*
Affiliation:
Carleton University, Ottawa, Ontario, Canada
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Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.

This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let gl,…,gn be a finite subset of G, and let

1—x is not a zero divisor so it is invertible and its inverse is 1+x+x3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 201 - 202
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Herstein, I. N., Notes from a ring theory conference, Amer. Math. Soc. 1971.Google Scholar
2. Kaplansky, I., “Problems in the theory of rings” revisited, Amer. Math. Monthly 77 (1970) 445-454. MR 41 #3510.Google Scholar