Some Researches On Clean Rings

In this thesis, we consider some rings such as nil. clean rings, strongly pseudo J-clean rings and strongly g(x)-J-clean rings.Firstly, it is proved that strongly nil clean rings are directly finite and the following questions are considered: (1)Let e be an idempotent. If both eRe and(1 - e)R(1 -e) are nil clean, whether is R nil clean or not? (2) If R is nil clean,whether are the corner rings of R weakly nil clean? For NR. rings, we give a positive answer about above questions.Secondly, some properties of potent elements were shown and the concept of strongly pseudo J-clean rings was defined. Some properties of strongly pseudo J-clean rings are given, and the following results were proved: (1) Strongly pseudo J-clean rings are strongly clean. (2) strongly pseudo J-clean rings are weakly abel,directly finite and have stable rauge one.Finally, we define the concept of strongly g(x)-J-clean rings and prove strongly J-clean rings are strongly x(x - 1)-J-clean. Some equivalent characteristics for strongly J-clean rings were given. The following results were proved: .R is a strongly (x-a.) (x-b)-J-clean ring if and only if R is a strongly J-clean ring and b - a + 1?J(R)if and only if R is a strongly J-clean ring and b - a - 1?J(R).