On clean, weakly clean, and feebly clean commutative group rings

Abstract

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. $R$ is said to be weakly clean if each element of $R$ is either a sum or a difference of a unit and an idempotent, and $R$ is said to be feebly clean if every element $r$ can be written as $r=u+e_1-e_2$, where $u$ is a unit and $e_1,e_2$ are orthogonal idempotents. Clearly clean rings are weakly clean rings and both of them are feebly clean. In a recent article (J. Algebra Appl. 17 (2018), 1850111(5 pages)), McGoven characterized when the group ring ${\mathbb{Z}}_{(p)}[C_q]$ is weakly clean and feebly clean, where $p,q$ are distinct primes. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, ${\mathcal{O}}_K$ its ring of integers, $\mathfrak{p}\subset \mathcal{O}$ a nonzero prime ideal, and ${\mathcal{O}}_{\mathfrak{p}}$ the localization of $\mathcal{O}$ at $\mathfrak{p}$. We investigate when the group ring ${\mathcal{O}}_{\mathfrak{p}}[G]$ is weakly clean and feebly clean, where $G$ 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 $K=\mathbb{Q}({\zeta }_n)$ is a cyclotomic field or $K=\mathbb{Q}(\sqrt{d})$ is a quadratic field.

To appear in Journal of algebra and its applications