Exchange Rings And Clean Rings

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.