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Trivial Units for Group Rings with $G$-adapted Coefficient Rings

Published:2005-03-01
Printed: Mar 2005
• Allen Herman
• Yuanlin Li
• M. M. Parmenter
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Abstract

For each finite group $G$ for which the integral group ring $\mathbb{Z}G$ has only trivial units, we give ring-theoretic conditions for a commutative ring $R$ under which the group ring $RG$ has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if $R$ is a ring of finite characteristic and $RG$ has only trivial units, then $G$ has order at most 3.
 MSC Classifications: 16S34 - Group rings [See also 20C05, 20C07], Laurent polynomial rings 16U60 - Units, groups of units 20C05 - Group rings of finite groups and their modules [See also 16S34]

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