On clean, weakly clean and feebly clean commutative group rings
On clean, weakly clean and feebly clean commutative group rings
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DOI：
10.1142/s0219498822500852
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发表时间：
2020-12
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影响因子：
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通讯作者：
Yuanlin Li;Qinghai Zhong
中科院分区：
文献类型：
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作者：
Yuanlin Li;Qinghai Zhong
如果R的每个元素可以写为单位的总和，则r r的环r被据称是弱的。如果每个元素可以写为r = u+ e1- e2，则dempotent且r被称为清洁，其中u是一个单位，e1，e2是正交的，显然是弱清洁的戒指。轻巧的清洁最近的文章（J. AlgebraAppl。17（2018），1850111（5页）），当组环Z（P）[CQ] [CQ]弱清洁且清洁时，McGoven的特征是P，Q在此中是不同的纸，我们考虑一个更通用的设置。是弱清洁且轻松清洁，其中G是有限的Abelian群体，并为这种组环的明确表征弱清洁，并且对于K = Q（ζn）是环形磁场或K = Q时的情况下是弱清洁且轻松清洁的（ √d）是一个二次字段。
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. R is said to be weakly clean if each element of R is either a sum or a difference of a unit and an idempotent, and R is said to be feebly clean if every element r can be written as r = u+ e1− e2, where u is a unit and e1, e2 are orthogonal idempotents. Clearly clean rings are weakly clean rings and both of them are feebly clean. In a recent article (J. Algebra Appl. 17 (2018), 1850111(5 pages)), McGoven characterized when the group ring Z(p)[Cq] is weakly clean and feebly clean, where p, q are distinct primes. In this paper, we consider a more general setting. Let K be an algebraic number field, OK its ring of integers, p ⊂ O a nonzero prime ideal, and Op the localization of O at p. We investigate when the group ring Op[G] is weakly clean and feebly clean, where G is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when K = Q(ζn) is a cyclotomic field or K = Q( √ d) is a quadratic field.