报告题目 (Title)：Quasi-clean Rings and Strongly Quasi-clean Rings（准清洁环与强准清洁环）

报告人 (Speaker)：唐高华 教授（北部湾大学）

报告时间 (Time)：2024年10月10 (周四) 15:30-17:00

报告地点 (Place)：校本部GJ303

邀请人(Inviter)：王卿文 教授

主办部门：理学院数学系

报告摘要： An element $a$ of a ring $R$ is called a quasi-idempotent if $a^{2}=k a$ for some central unit $k$ of $R$, or equivalently, $a=k e$, 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 $\mathbb{Z}_{2}$; For an indecomposable commutative semilocal ring $R$ with at least two maximal ideals, $\mathbb{M}_{n}(R)(n \geq 2)$ is strongly quasi-clean if and only if $\mathbb{M}_{n}(R)$ 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 $n \geq 2$, $\mathbb{M}_{n}\left(\mathbb{Z}_{(p)}\right)$ is strongly quasi-clean if and only if $p>n$. Some open questions are also posed.