Certain Rings Related To Jacobson Radicals

This dissertation is mainly concerned on clean rings,we make certain exten-sions of its structure and make further studies on certain rings related to Jacobson radicals.The nature of J-clean rings has been studies in a more detained way and then such new types of rings has been gained.The innovativeness of the thesis chiefly comprises:(1)The paper has extended pseudo weakly clean rings to pseu-do weakly J-clean rings and give certain descriptions of the characterizations of pseudo weakly J-clean rings.(2)Based on the definition of J-clean rings,the more elaborate research about J-clean rings has been made.(3)Then the research has extended UR-rings to JR-rings and got various properties of JR-rings;On the basis of the above study,the properties,which are about the above three kinds of rings,has been obtained.This paper contains the following parts:The first part:The background of clean rings and their developments are introduced,and the main research of this paper is also presented.The second part:The elementary definitions involved in this thesis and rele-vant important conclusions are listed.The third part:According to the definition of pseudo weakly clean rings,We extend J-clean rings to pseudo weakly J-clean rings.We investigate various properties of pseudo weakly J-clean rings;and explore the relationship between pseudo weakly J-clean rings and its related rings.Further,we study conditions under which pseudo weakly J-clean rings and its related rings are equivalent.The fourth part:Further study on the properties of J-clean and Morita con-texts.we prove that:Let R:=(A M N B)be a Morita context.Then R is J-clean ring if and only if A,B are J-clean rings and MN(?)J(A)and NM(?)J(B);Let R be a ring with s(?)C(R),then the generalized matrix ring Ks(R)is J-clean if and only if R is J-clean and s ∈ J(R);Let R be a ring with s ∈ C(R),then Mn(R;s)is J-clean if and only if R is J-clean and s ∈ J(R).The fifth part:We introduce JR-rings,and then extend the known results of UR-rings to JR-rings.We get many useful properties and prove that R a JR-ring if and only if R/J(R)is regular and regular elements lift modulo J(R).A ring R is a Boolean ring if and only if every element in R is the unique sum of a regular element and an element in J(R).Further,we investigate the behavior of these properties under various ring extensions.Finally,the dissertation puts forward the further possible research which is related to the topics in this paper.