正则性及广义逆理论

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.

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。

环的正则性与元素的广义逆紧密相关，比如*-正则环（强正则环，强∏-正则环）等价于每一个元素有MP逆（群逆，Drazin逆）。本项目讨论了正则性与各种广义逆问题，主要研究了：（1）*-正则环，*-幺正则环，*-强正则环及与MP逆，群逆，Drazin逆，核逆的关系。（2）MP逆，群逆，Drazin逆，广义Drazin逆和伪Drazin逆的存在性及表达式；（3）几类新型广义逆（核逆与对偶核逆，相对于一个元素的逆，（b,c）逆）的存在性及表达式；（4）Banach代数，C*-代数中各种广义逆的性质和刻画；（5）环的clean性，*-clean性及拟（伪）polar性。本项目采用了环与模理论，范畴论，算子理论等工具和方法，对正则性和广义逆展开研究，在SCI期刊上正式发表 45篇论文，在线发表6篇论文。这些成果对理清正则性与广义逆的关系是非常重要的，同时经典广义逆（MP逆，群逆，Drazin逆）的新结果丰富了广义逆理论，新型广义逆（核逆与对偶核逆，相对于一个元素的逆，（b,c）-逆）的结果是新的，完善了正则性及广义逆理论，并可应用于Banach代数与C*-代数中去。

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