【作者】 张振华；

【导师】 肖光世；

【作者基本信息】 南京航空航天大学 ， 基础数学， 2010， 硕士

【摘要】 自从Ehrlich提出单位正则环的概念以来,国内外许多代数工作者对其进行了深入研究.随后,Warfield,Nicholson分别将单位正则环推广到exchange环,clean环.肖光世和佟文廷进一步将clean环推广到n - clean环.本文主要讨论exchange环和clean环.首先,讨论clean环和2 - clean环的性质.其次,讨论n - clean一般环的幂级数环(包括左斜幂级数和右斜幂级数环)和矩阵环的性质,并且定义了强n - clean一般环,讨论强n - clean一般环的矩阵环和它的扩张.最后,讨论clean环,2 - clean环与exchange环关系.通过减弱“R是左拟- duo环”的条件,分别在“R是左(右)弱理想,广义弱理想,拟理想,补理想”的条件下来讨论clean环与exchange环的关系,得到了如果R是左(右)弱理想,广义弱理想,拟理想或补理想,那么R是clean环当且仅当R是exchange环.并将所得结果推广到2 - clean环.更多还原

【Abstract】 Since the unit-regular ring conception was introduced by Ehrlich, many algebraic scholars have studied it. Later, it was extended to the exchange ring and the clean ring by Warfield and Nicholson. Later, the n-clean ring was introduced by Xiao Guangshi and Tong Wenting.The main purpose of this paper is to discuss exchange rings and clean rings. Firstly, we introduced some properties of clean rings and 2-clean rings. Seconddly, we discussed n-clean power series rings (including left power series rings and right power series rings) and n-clean matrix rings. We also introduce strong n-clean general rings and some properties, such as the matrix and extention of strong n-clean rings. At the end of the paper, we study the relationship between clean rings, 2-clean rings and exchange rings. By weakening the condition that“R is a quai-duo ring”, some conditions are proved. If“every maximal left ideal of R is a GW-ideal”,“every maximal left ideal of R is a quasi-ideal”,“every maximal left ideal of R is a weakly right ideal”,“every complement left ideal of R is an ideal”, we obtain that R is exchange if and only if it is clean. We also extend the results to 2-clean rings.更多还原