Clean group rings over localizations of rings of integers

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. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring ${\mathbb{Z}}_{(p)}[C_n]$ is clean, where ${\mathbb{Z}}_{(p)}$ is the localization of the integers at the prime $p$. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, ${\mathcal{O}}_K$ be its ring of integers, and $R$ be a localization of ${\mathcal{O}}_K$ at some prime ideal. We investigate when $R[G]$ is clean, where $G$ is a finite abelian group, and obtain a complete characterization for such a group ring to be 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 Pure and Applied Algebra