关于clean环与exchange环的一点注记  

A note on clean ring and exchange ring

作　　者：张振华[1] 肖光世[1] 

机构地区：[1]南京航空航天大学理学院,江苏南京211106

出　　处：《宝鸡文理学院学报（自然科学版）》2010年第1期6-7,10,共3页Journal of Baoji University of Arts and Sciences(Natural Science Edition)

摘　　要：目的研究clean环和exchange环的关系。方法环R被称为clean环,若环R中每一个元素都能写成一个幂等元和一个单位的和。通过减弱了"R是左拟-duo环"的条件,来研究clean环和exchange环的关系。结果证明了,如果"R的任意极大左理想是GW-理想"、"R的任意极大左理想是拟理想"、"R的任意极大左理想是弱右理想"以及"R的任意补左理想是理想",那么R是exchange环当且仅当R是clean环。结论在减弱"R是左拟-duo环"得到的一些条件下,R是exchange环当且仅当R是clean环成立。Aim To investigate the relationship between clean ring and exchange ring. Methods A ringR is said to be clean if every element can be written as a sum of an idempotent and a unit. The relationship between clean ring and exchange ring is investigated by weakening the condition that R is a left quasi-duo ring. Results 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, R is exchange if and only if it is clean. Conclusion It is true that R is an exchange ring if and only if it is clean, by weakening the condition that R is a left quasi-duo ring.

关 键 词：CLEAN环 EXCHANGE环 GW-理想 拟理想 弱理想 

分 类 号：O153.3[理学—数学]