稳定度条件与环的正则性、clean性

Stable range conditions originated from the research of general linear groups in algebraic K-theory, which act important role in ring theory, module theory, algebraic K-theory and the other related fields, and connect a lot of conceptions of Algebra such as K-groups, the direct summand cancellation, substitution, exchange property, regularity and cleanness of rings. Along the main route stable range conditions associating with the regularity and cleanness of rings, this project will investigate: (1) properties and connections of related stable range conditions; (2) related stable range conditions and cleanness of polynomial rings, power series rings, matrix rings, group rings and continious functional rings; (3) relationship between related stable range conditions and, regularity and cleanness of rings; (4) (power) cancellation and (power) subsititution of modules satisfying related stable range conditions; (5) K-group structure of rings satisfying related stable range conditions, and further giving new characterizations of regularity and cleanness of rings. The results obtained will improve to learn deeply stable range conditions and, regularity and cleanness of rings.

稳定度条件起源于代数K-理论中一般线性群的研究，在环论、模论、代数K-理论及相关领域中扮演着重要的角色，关联着代数学中的诸多概念，如K群，模的直和可消性、substitution性、exchange性以及环的正则性和clean性等。本项目拟将以相关稳定度条件为主线，结合环的正则性和clean性研究：（1）相关稳定度条件的性质及其之间的内在联系；（2）多项式环、幂级数环、矩阵环、群环和连续函数环等扩张环的相关稳定度条件和clean性；（3）相关稳定度条件与环的正则性、clean性之间的内在联系，利用环的相关稳定度条件刻画环的正则性和clean性；（4）具有相关稳定度条件模的（幂）直和可消性、（幂）substitution性；（5）满足相关稳定度条件环的K群结构，进而刻画环的正则性、clean性。最终的研究成果有望深化人们对稳定度条件及环的正则性、clean性的认识。

稳定度条件起源于代数K-理论中一般线性群的研究，在环论、模论、代数K-理论及相关领域中扮演着重要的角色，关联着代数学中的诸多概念，如K 群，模的直和可消性、substitution 性、exchange 性以及环的正则性和clean 性等。本项目结合环的稳定度条件、正则性，研究了环的相关clean性，得到了如下主要结果：（1）建立了幺正则环，强π-正则环、J-拟polar环、伪polar环、*-clean环及强nil-clean环与稳定度1条件、幂等稳定度1条件和投影稳定度1条件之间的关系；（2）给出矩阵环是J-拟polar的充分必要条件，得到*-clean群环的充分和必要条件；（3）得到强π-正则环下，环有单位2和的条件；在稳定度1条件下，得到只要1有单位2和，则环中所有正则元有单位2和；从而得到幺正则环条件下，环有单位2和的充分必要条件是1有单位2和；（4） 在环中研究伪polar性，得到其与正则性、稳定度条件和强clean性之间的关系；建立了伪polar环与伪Drazin逆的联系，得到Banach代数中两个元素和的伪Drazin公式；（5）结合环的强π-正则性和强clean性，研究了强nil-clean环，给出了其结构定理（包括一般环的情形），给出群环强nil-clean的充分和必要条件，通过环的强nil-clean性给出矩阵环是nil-clean的条件，推广了Diesl，Breaz和Zhou等人的工作。