正则环是环论中重要的研究对象，广义逆（Moore-Penrose逆、Drazin 逆）与正则环（*-正则环、强pi-正则环）有深刻的联系，在许多领域有重要应用。随着研究的深入，相继出现了广义（伪）Drazin逆及与之对应的拟（伪）polar环，为广义逆理论的研究注入了新的活力。本项目的主要研究内容有：（1）环上元素和矩阵的（广义）MP-逆、(b, c)-逆及*-（强/幺）正则环的刻画；（2）环上元素及矩阵的（广义/伪）Drazin 逆与（拟/伪）polar环的刻画；（3）环的正则性与相关clean性、拟（伪）polar性之间的联系；（4）广义逆在C*-代数、Banach代数中的应用。这些内容的研究一方面丰富环的正则性理论，另一方面将复矩阵、有界线性算子及Banach代数上的广义逆理论推广到更一般的情形，为C*-代数和Banach 代数的分类和谱理论研究提供新思路。

Regular ring is an important object researched in ring theory. Generalized inverse (Moore-Penrose inverse, Drazin inverse) is deeply related to regular ring (*-regular ring, strongly pi-regular ring) and plays an important role in many area. In the course of deepening the research, the notion of generalized (pseudo) Drazin inverse and, accordingly, the notion of quasi (pseudo) polar ring are introduced so that the theory of generalized inverse becomes more flourishing. The main contents of this project are as follows. (1) Characterizations of (generalized) MP-inverse and (b, c)-inverse of elements and matrixes over rings and characterizations of *-(strongly, unit) regular rings; (2) Characterizations of (generalized/pseudo) Drazin inverse of elements and matrices over rings and characterizations of (quasi/pseudo) polar rings; (3) Relationship among regularity, relative cleanness and quasi (pseudo) polarity of rings; (4) Applications of generalized inverse in C*-algebras and Banach algebras. The investigation into these contents will enrich the theory of regularity of rings and extend the theory on generalized inverse of complex matrices, bounded linear operators and elements in Banach algebras to more generalized cases. This will also provide some new idea for the classification and the spectra theory of C*-algebras and Banach algebras.