Clean group rings over localizations of rings of integers

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Li, Yuanlin^1,2  Zhong, Qinghai³ 

[1] Brock Univ, Dept Math & Stat, 1812 Sir Isaac Brock Way, St Catharines, ON L2S 3A1, Canada

[2] Jiangsu Univ, Fac Sci, Zhenjiang, Jiangsu, Peoples R China

[3] Karl Franzens Univ Graz, Inst Math & Sci Comp, NAWI Graz, Heinrichstr 36, A-8010 Graz, Austria

关键词: Clean ring;    Group ring;    Ring of algebraic integers;    Primitive root of unity;    Cyclotomic field;   

DOI  :  10.1016/j.jpaa.2019.106284

来源: Elsevier

【 摘 要 】

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))[C-n] 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, O-K be its ring of integers, and R be a localization of 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 = Q(zeta(n)) is a cyclotomic field or K = Q(root d) is a quadratic field. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free