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 Z(p)[Cn] is clean, where 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, OK be its ring of integers, and R be a localization of OK 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=Q(ζn) is a cyclotomic field or K=Q(d) is a quadratic field.