拟polar环与广义Drazin逆

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。

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.