有对合的零清洁环

IF 0.4 4区数学 Q4 MATHEMATICS

Algebra Colloquium Pub Date : 2021-07-26 DOI:10.1142/s1005386721000298

Jian Cui, Guoli Xia, Yiqiang Zhou

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引用次数: 1

摘要

如果[公式:见文本]的每个元素都是投影和幂零的和，则一个[公式:见文本]-环[公式:见文本]被称为nil[公式:见文本]-干净环。Nil[公式:见文本]-clean rings是[公式:见文本]- diesel引入的Nil -clean rings的版本。本文讨论了环的零[公式:见文]-洁净性，重点讨论了矩阵环。我们证明了[公式:见文]-环[公式:见文]为nil[公式:见文]-清洁当且仅当[公式:见文]为nil且[公式:见文]为nil[公式:见文]-清洁。对于一个2-原[公式:见文]-环[公式:见文]，在[公式:见文]给出的诱导对合下，[公式:见文]的零[公式:见文]-洁净性质完全化约为[公式:见文]的零[公式:见文]性质。因此，[Formula: see text]不是nil [Formula: see text]-clean ring for [Formula: see text]， [Formula: see text]是nil [Formula: see text]-clean ring当且仅当[Formula: see text]为nil， [Formula: see text]是布尔环，[Formula: see text]为所有[Formula: see text]。

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Nil-Clean Rings with Involution

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

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来源期刊

Algebra Colloquium 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

625

审稿时长

15.6 months

期刊介绍： Algebra Colloquium is an international mathematical journal founded at the beginning of 1994. It is edited by the Academy of Mathematics & Systems Science, Chinese Academy of Sciences, jointly with Suzhou University, and published quarterly in English in every March, June, September and December. Algebra Colloquium carries original research articles of high level in the field of pure and applied algebra. Papers from related areas which have applications to algebra are also considered for publication. This journal aims to reflect the latest developments in algebra and promote international academic exchanges.