环的clean性及其相关广义逆的研究

Cleanness of rings grew out of the study of exchange rings which act an important role in the cancellation of modules, and connect a lot of conceptions in algebras (such as, stable range condition, regularity). In recent years, some new classes of clean rings that are derived from relative subjects relate ring theory to generalized inverse theory and C*-algebra deeply. This project aims to make research from the following aspects. From the whole point of view, the structures, properties and connections of related clean rings will be investigated; stable range conditions, extension properties and K-group structures of rings such as strongly clean rings and quasipolar (pseudopolar) rings will be concerned; and the class of clean rings will be reclassified by introducing the notion of "index". In addition, we will study *-cleanness and *-regularity of C*-algebras associating with the regularity and cleanness of rings, establish the theories of *-clean rings and *-regular rings. From the local perspective, we will investigate EP elements, quasinilpotent elements and its relationship to the Jacobson radical in a ring, and consider the existences and applications of (generalized) Drazin inverses, pseudo Drazin inverses and Moore-Penrose inverses in both rings and Banach algebras. The final work is expected to make some new progress on open questions of related clean rings, and the results obtained will improve to learn deeply the cleanness of rings and the related generalized inverses.

环的clean性起源于在模消去理论中扮演重要角色的exchange环的研究，关联着代数学中的诸多概念(如稳定度条件、正则性)。近年来，从相关领域派生出一些新的clean环将环论与广义逆理论、C*-代数更加紧密地联系起来。本项目拟从如下角度展开：整体上研究相关clean环的结构、性质及内在联系，考虑诸如强clean环、拟(伪)polar环的稳定度条件、扩张性质和K群结构，引入“指标”概念对clean环进行重新分类，结合正则性、clean性探讨C*-代数的*-clean性、*-正则性，完善*-clean环、*-正则环理论；局部上研究EP元、拟幂零元及其与Jacobson根的关系，寻求环及Banach代数中元素(广义)Drazin逆、伪Drazin逆、Moore-Penrose逆的存在条件及应用。最终有望在关于clean环的公开问题上取得新进展，并深化人们对环的clean性及其相关广义逆的认识。

环的clean性来源于exchange环的研究，不仅关联着正则性、稳定度条件等诸多概念，而且与C*-代数、广义逆理论密切相关。本项目主要围绕环的clean性及其相关广义逆展开研究，研究内容和结果主要包括：(1) 研究了（强）*-clean环的性质、结构及其投影稳定度条件，构造了许多例子，部分解决了关于强clean环的若干公开问题；(2) 结合MP-逆研究（幺正则）*-正则环的结构，通过k-GN性质给出矩阵环具有（幺正则）*-正则性的充分必要条件；(3) 把π-正则性拓展到*-环，引入并研究π-*-正则环的性质，给出了阿贝尔的π-*-正则环的结构定理；(4) 研究了诣零*-clean环与n-*-clean环，建立了它们与相关*-clean环的内在联系；(5) 引入强拟诣零clean（元）环，研究了强拟诣零clean环与拟polar环、强诣零clean环等环类的关系，证明了强拟诣零clean元对应着一类特殊的广义Drazin逆；(6) 研究了伪polar环在指数为一时的结构、性质，进一步研究了强J-clean环及半clean群环，推广了许多已知重要结论。. 本项目的研究成果丰富和发展了环的clean性及相关广义逆的理论。在项目资助下，项目组在国内外重要学术期刊发表和录用研究论文11篇，其中4篇属于SCIE期刊。项目实施过程中，项目组成员积极参加学术交流；注重研究生培养，共毕业研究生4人，其中1人考取南京大学博士研究生；1名成员晋升为副教授。