拟polar环与广义Drazin逆

The quasipolarity of rings originated from generalized inverse theory and spectral theory in Banach algebras, which is closely related to generalized Drazin inverses. Meanwhile, quasipolar rings erect a bridge between strongly pi-regular rings and strongly clean rings. In recent years, the study of cleanness of rings has been a hotspot in ring theory. From the whole point of view, this project will investigate the structures of quasipolar rings, establish the connections between quasipolar rings and the related clean rings, investigate their stable range conditions and directly finite properties, consider open problems on strongly clean rings by means of generalized inverses; and from the local perspective, we study the properties of a quasipolar element, find out its relations with related clean elements, and discuss existences of Drazin inverses and generalized Drazin inverses in rings. Consequently, the related results obtained will be used in Banach algebras and bounded linear operators over Banach spaces. The research on quasipolar rings and generalized Drazin inverses not only connects deeply the theory of rings with the theory of generalized inverses, but also brings new ideas to the research of cleanness of rings.

环的拟polar性起源于Banach代数中的广义逆理论和谱理论，与广义Drazin逆密切相关；同时拟polar环是联系强pi-正则环和强clean环的桥梁。近年来，环的相关clean性是环论研究的一个热点。本项目拟从整体上研究拟polar环的结构，构建它与相关clean环的内在联系，探讨其稳定度条件、直有限性问题，借助广义逆研究强clean环中的公开问题；局部上研究拟polar元的性质，弄清它与相关clean元的关系，研究环中元素Drazin逆、广义Drazin的存在性并将相关结果应用到Banach代数和Banach空间有界线性算子中。拟polar环和广义Drazin逆的研究不仅会深化环论和广义逆理论的联系，而且能为环的clean性研究提供新思路。

环的拟polar性与广义Drazin逆紧密相关，同时拟polar环严格介于强pi-正则环与强clean环之间。环的相关clean性是近年来环论研究的一个重要组成部分。本项目进一步研究了拟polar环的性质、结构；借助友矩阵给出了非交换局部环上矩阵环的拟polar性判定准则，并刻画了拟polar矩阵与强clean矩阵的内在关系；研究了环中元素广义Drazin逆的存在性，得到了关于广义Drazin逆的Cline公式。引入了强pi-*-正则环的概念，建立了它与（强）pi-正则环、强*-clean环、强*-正则环等重要环类的内在联系；此外把环的强clean性的相关研究结果应用到线性代数中。基于上述内容，项目申请者已发表3篇论文（包括1篇教研论文），另有1篇论文在审稿中。