Quasi-clean rings and strongly quasi-clean rings

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:唐高华(北部湾大学)
:2023-09-17 14:30
:海韵园实验楼105报告厅

报告人:唐高华北部湾大学

 间:202391714:30

 点:海韵园实验楼105报告厅

内容摘要:

An element a of a ring R is called a quasi-idempotent if a2 = ka for some central unit k of R, or equivalently, a = ke, where k is a central unit and e is an idempotent of R. A ring R is called a quasi-Boolean ring if every element of R is quasi-idempotent. A ring R is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or R has no image isomorphic to Z2; For an indecomposable commutative semilocal ring R with at least two maximal ideals, Mn(R)(n ≥ 2) is strongly quasi-clean if and only if Mn(R) is quasi-clean if and only if min{|R/m|, m is a maximal ideal of R} > n + 1. For a prime p and a positive integer n ≥ 2, Mn(Z(p)) is strongly quasi-clean if and only if p > n. Some open questions are also posed.

人简介

唐高华,北部湾大学教授,博士生导师,广西十百千人才,全国优秀教师,八桂名师,广西高校教学名师,教育部高等学校数学类专业教学指导委员会委员,广西高校数学类专业教学指导委员会主任委员,广西数学会理事长。主要从事交换代数、同调代数、环的代数结构与图结构、形式矩阵环等的研究。定义了交换环的弱Krull维数,证明了弱Krull维数为2的广义伞环上Bass-Quillen猜想成立。建立了环上形式矩阵环理论,其中的一类被称之为唐-周环。在环的内部刻画、环的同调理论、环的代数结构与图结构、环上形式矩阵环等的研究中取得了系列成果,发表论文150多篇。

 

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