On Weakly J-Clean And Unit J-Clean Rings

This dissertation is mainly composed of J-clean rings, we make certain exten-sions of J-clean rings and make further studies on the properties of J-cleanness. We explore the relations between J-clean rings and its related rings such as clean rings in details in terms of such new types of rings. The innovative job of this thesis mainly includes:(1) We extend J-clean rings to weakly J-clean rings and get various properties of weakly J-clean rings; (2) We introduce unit J-clean rings and give certain descriptions of the characterizations of unit J-clean rings. The structures of J-clean rings and related rings are thereby investigated. This paper contains the following parts:The first part:We introduce the background of J-clean rings as well as their developments. The main research of this paper is list as well.The second part:We introduce the elementary definitions involved in this thesis and the important conclusions related to this paper are also listed.The third part:We introduce weakly J-clean rings. We study the properties of such new type of rings, which extend many known results on J-clean rings. We investigate various properties of weakly J-clean rings; and explore the relation-ship between weakly J-clean rings and its related rings. Further, we also study conditions under which weakly J-clean rings and its related rings are equivalent.The fourth part:We introduce unit J-clean rings, and then extend the known results of J-clean rings to weakly J-clean rings. We get many useful properties, such as every J-clean rings is unit J-clean rings; every unit J-clean rings is unit clean rings; every 2-good rings is unit clean rings and so on. In particular, we shall characterize such rings by means of the lifting property.Finally, we put forward the further possible research related to our topics in this article.