On Idempotent Units in Commutative Group Rings

Abstract

Special elements as units, which are defined utilizing idempotent elements, have a very crucial place in a commutative group ring. As a remark, we note that an element is said to be idempotent if r2 = r in a ring. For a group ring RG, idempotent units are defined as finite linear combinations of elements of G over the idempotent elements in R or formally, idempotent units can be stated as of the form id(RG) = {∑rg ∈id(R) rg g: ∑rg ∈id(R) rgg = 1 and rg rh = 0 when g ≠ ℎ} where id(R) is the set of all idempotent elements [3], [4], [5], [6]. Danchev [3] introduced some necessary and sufficient conditions for all the normalized units are to be idempotent units for groups of orders 2 and 3. In this study, by considering some restrictions, we investigate necessary and sufficient conditions for equalities: i. V(R(G × H)) = id(R(G × H)), ii. V(R(G × H)) = G × id(RH), iii. V(R(G × H)) = id(RG) × H where G × H is the direct product of groups G and H. Therefore, the study can be seen as a generalization of [3], [4]. Notations mostly follow [12], [13]. © 2020, Sakarya University. All rights reserved.