On clean, weakly clean and feebly clean commutative group rings

Abstract

A ring [Formula: see text] is said to be clean if each element of [Formula: see text] can be written as the sum of a unit and an idempotent. [Formula: see text] is said to be weakly clean if each element of [Formula: see text] is either a sum or a difference of a unit and an idempotent, and [Formula: see text] is said to be feebly clean if every element [Formula: see text] can be written as [Formula: see text], where [Formula: see text] is a unit and [Formula: see text] are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] are distinct primes. In this paper, we consider a more general setting. Let [Formula: see text] be an algebraic number field, [Formula: see text] its ring of integers, [Formula: see text] a nonzero prime ideal and [Formula: see text] the localization of [Formula: see text] at [Formula: see text]. We investigate when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when [Formula: see text] is a cyclotomic field or [Formula: see text] is a quadratic field.