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].