稳定度条件与环的正则性、clean性

项目名称： 稳定度条件与环的正则性、clean性

项目编号： No.11201064

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王周

作者单位： 东南大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 稳定度；正则性；clean 性；群环；强Drazin 逆

英文摘要： 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.

英文关键词： stable range；regularity；cleanness；group rings；strongly Drazin inverses