Trivial units in commutative group rings of $G\times C_{n}$

Abstract

It is known that if the unit group of an integral group ring $\mathbb{Z}G$ is trivial, then the unit group of $\mathbb{Z}(G\times C_{2})$ 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 $D=G\times H$ of abelian groups $G$ and $H$, such that the unit group of the ring $R(G\times H)$ 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 $G\times C_{3}$, $G\times K_{4}$ or $G\times C_{4}$, 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].