报 告 人：Xavier Mary 教授

工作单位：Université Paris-Ouest Nanterre – La Défense

举办单位：数学学院

报告人简介： Xavier Mary, 2006年博士毕业于巴黎大学。目前为法国巴黎第十大学教授，主要研究领域包括：半群理论、环论、代数等。X. Mary 教授在国际代数领域享有很高的知名度，于2011年引入了“The inverse along an element”,现称之为 Mary 逆，被广泛的研究。目前国际上以Mary 逆命名此逆。目前，X. Mary 教授已在 Linear Algebra Appl., Linear Multilinear Algebra, Appl. Math. Comput., Comm. Algebra  J. Algebra Appl.等杂志上发表了30篇学术论文。并且，X. Mary 教授主持过欧洲地平线2020项目。

一、IC rings and transitivity of perspectivity

报告时间：2022年12月24日（星期六）14:00-15:00

报告地点：腾讯会议：991-474-798

报告简介：We construct an example of an IC ring where perspectivity is transitive, but not all isomorphic idempotents are perspective. We also develop new criteria for checking perspectivity of idempotents in rings.

二、Special clean elements, perspective elements and perspective rings

报告时间： 2022年12月26日（星期一）14:00-15:00

报告地点：腾讯会议：991-474-798

报告简介：Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of a ring. We notably prove that perspective elements form a proper subset of special clean elements, and that perspectivity of elements is a left-right symmetric property. An equational characterization of perspective elements is also given and examples are provided.

三、N-chained semigroups and n/2-perspective modules and rings

报告时间：2022年12月28日（星期三）14:00-15:00

报告地点：腾讯会议：991-474-798

报告简介：We introduce and study a new class of modules and rings we call n/2-perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small (≦2), we recover existing class of modules and rings: (endo)abelian, strongly IC, and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.

四、Characterizations of clean elements by means of outer inverses in rings and applications

报告时间：2022年12月31日（星期六）14:00-15:00

报告地点：腾讯会议：496-349-969

报告简介：We characterize clean elements in unital and general rings by means of outer inverses. Some special cases, such as both clean and unit-regular elements, or strongly clean elements, are discussed. As an application, we also derive new characterizations of strongly regular elements.

五、Characterizations of special clean elements and applications

报告时间：2023年1月3日（星期二）14:00-15:00

报告地点：腾讯会议：506-441-906

报告简介：We prove that special clean decompositions of a given element of a ring are in one-to-one correspondence with the set of solutions of a simple equation in a corner ring. We then derive “constructive” proofs that in many rings, regular elements are special clean by solving this equation in specific cases. Other applications, such as uniqueness of decompositions, are given. Many examples of special clean decompositions of 2-2 matrices found by this methodology are also presented.