The Structures Of Certain ClEAN Rings

In this thesis, we investigate certain subclasses of clean rings, and extend many known clean properties to a wider type of rings. It includes three new top-ics:extend the properties of quasipolar rings to weakly quasipolar rings; extend the properties of nil-quasipolar rings to weakly nil-quasipolar rings, and then in-vestigate structures of such rings; furthermore, we combine quasipolar properties to strong J-cleanness, and introduce many properties of strongly J-clean rings are thereby extended to such larger class of clean rings. These generalizations have better properties and more widely usage than the known ones. This paper includes the following parts:The first part:we introduce the background of clean rings as well as their developments. The main research of this paper is as well list;The second part:we introduce the elementary definitions involved in this paper and the important conclusions related to this paper are also list;The third part:we introduce weakly quasipolar rings. We study the proper-ties of weakly quasipolar rings, which extended many known results on quasipolar rings. The equivalent conditions of weakly quasipolar rings and quasipolar rings are given. It is proved that a strongly π-regular ring is a weakly quasipolar ring, and a weakly quasipolar ring is a strongly clean ring. Further, we discuss the weakly quasipolarty about module endomorphism rings. The weak quasipolartity about triangular matrix rings over a local ring is also explored.The fourth part:we introduce weakly nil-quasipolar rings, and then extend the known results of nil-quasipolar rings to weakly nil-quasipolar rings. It is proved that a weakly nil-quasipolar ring and a strongly nil-clean ring coincides with each other. The conditions under which a 2 x 2 diagonal matrix is weakly nil-quasipolar are obtained.The fifth part:we introduce strongly J1/2-clean rings. The properties of strong-lv J-clean rings are generalized to stronglv J1/2-clean rings. The equivalence of strong J1/2-cleanness and the strong J1/2-cleanness on triangular matrices over a ocal ring are established.Finally, we put forward the further possible research related to our topics in this article.