2019/11/12 by Li, Yuanlin, Zhong, Qinghai
#11R11 #11R18 #16S34 #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1911.04713
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)[Cn] 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 OK be its ring of integers, and R be a localization of \mathcal 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=\mathbb Q(ζn) is a cyclotomic field or K=\mathbb Q(√(d)) is a quadratic field.