Clean group rings over localizations of rings of integers

TL;DR

This paper characterizes when group rings over localizations of rings of integers in number fields are clean, extending previous results to more general algebraic number fields and specific classes like cyclotomic and quadratic fields.

Contribution

It provides a complete characterization of the cleanness of group rings over localizations of rings of integers in algebraic number fields, generalizing earlier work on integers.

Findings

Characterization for cyclotomic fields

Results for quadratic fields

Extension of previous cleanness criteria

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.