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 𝑟 ଶ = 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) = {∑ 𝑟௚𝑔 ೒∈௜ௗ(ோ) : ∑೒∈௜ௗ(ோ) 𝑟௚ = 1 and 𝑟௚𝑟௛ = 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. 𝑉൫𝑅(𝐺 × 𝐻)൯ = 𝑖𝑑(𝑅(𝐺 × 𝐻)), 𝑖𝑖. 𝑉൫𝑅(𝐺 × 𝐻)൯ = 𝐺 × 𝑖𝑑(𝑅𝐻), 𝑖𝑖𝑖. 𝑉൫𝑅(𝐺 × 𝐻)൯ = 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]. Keywords: idempotent, unit, group ring, commutative