Trivial Units in Commutative Group Rings of G X Cn

Abstract

It is known that if the unit group of an integral group ring ZG is trivial, then the unit group of Z(G x C-2) is trivial as well [3]. The aim of this study is twofold: firstly, to identify rings R that are D-adapted for the direct product D = G x H of abelian groups G and H, such that the unit group of the ring R(G x H) is trivial. Our second objective is to investigate the necessary and sufficient conditions on both the ring R and the direct factors of D to satisfy the property that the normalized unit group V (RD) is trivial in the case where D is one of the groups G x C-3, G x K-4 or G x C-4, where G is an arbitrary finite abelian group, C-n denotes a cyclic group of order n and K-4 is Klein 4-group. Hence, the study extends the related result in [18].