Moore-Penrose逆、Drazin逆是非常重要的两类广义逆，在复矩阵、Banach代数、C*-代数、环论等相关领域中有着重要的应用。Moore-Penrose逆、Drazin逆分别与*-正则元、强pi-正则元有着密切的联系。近年来, 几类新引入的广义逆为其发展注入了新的活力。环的clean性在环模理论、C*-代数和广义逆理论中扮演着重要的角色。本项目拟对环上元素的广义逆及其正则性、clean性进行研究。主要内容包括：建立环中元素的广义逆与正则性、clean性的联系, 并将所得结果对环进行分类。环上几类广义逆之间的联系及其存在性刻画。环中元素的e-核逆、Mary逆及混合(b,c)-逆的性质与计算，并将相关结果应用到Banach代数与C*-代数中。

Moore-Penrose inverses and Drazin inverses are two types of important generalized inverses, which have been widely used in the fields of complex matrices, Banach algebras, C*-algebras, rings and so on. Moore-Penrose inverses and Drazin inverses are closely related to *-regular elements and strongly pi-regular elements, respectively. Recently, several types of new introduced generalized inverses bring vitality to the development of the theory of generalized inverses. The cleanness of rings plays an important role in the theory of the ring-module, the C*-algebra and the theory of generalized inverses in rings. This project devotes to the research of generalized inverses of elements in rings and their regularites and cleannesses. The main contents include: The connection between generalized inverses of elements of rings and their regularities and cleannesses, and the obtained results will be used to classify rings. Characterizations and relations among several new types of generalized inverses in rings. The property and the computation of e-core inverses, Mary inverses and hybrid (b,c)-inverses. Also, the related results of generalized inverses will be applied to Banach algebras and C*-algebras.