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A NOTE ON GROUP RINGS WITH TRIVIAL UNITS

Part of: Rings and algebras arising under various constructions Conditions on elements

Published online by Cambridge University Press:  19 July 2021

A. Y. M. CHIN Open the ORCID record for A. Y. M. CHIN [Opens in a new window]
A. Y. M. CHIN*
Affiliation:
Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603Kuala Lumpur, Malaysia
*
e-mail: acym@um.edu.my
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Abstract

Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

Keywords

group ringunittorsion grouplocally finite group

MSC classification

Primary: 16S34: Group rings , Laurent polynomial rings
Secondary: 16U60: Units, groups of units
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 243 - 247
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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