题目: Quasi-clean rings and strongly quasi-clean rings
摘要: An element a of a ring R is called a quasi-idempotent if a^2 = 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 Z_2; For an indecomposable commutative semilocal ring R with at least two maximal ideals, M_n(R) (n>1) is strongly quasi-clean if and only if M_n(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>1, M_n(Z_(p)) is strongly quasi-clean if and only if p > n. Some open questions are also posed.
报告人: 唐高华
报告时间：2024年5月13日 上午10：00-11：00
报告地点：西南交通大学犀浦校区第7号教学7510
报告人简介：唐高华，北部湾大学党委副书记，二级教授，博士生导师，广西十百千人才，全国优秀教师，全国师德先进个人，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。主要从事数学、数学教育和高等教育国际化的研究。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。主持和承担国家自然科学基金项目7项，省部级项目10多项，发表论文150多篇，获得省级科技进步奖1项，省部级优秀教学成果奖5项。
