The Structures Of Certain Quasipolar Rings

This dissertation is mainly concerned on J-quasipolar rings,the properties of the J-quasipolar rings are studied in more detail.We make certain extensions of its structure and make further studies on certain rings related to Jacobson radicals,then a new type of rings has been gained.The innovative work of the paper mainly includes:(1)Optimized the definition of the J-quasipolar rings,the properties of the J-quasipolar rings are studied in more detail.(2)A new type of ring is depicted,and its structure and properties are studied.(3)We investigated the structure theorems of 3 x 3 matrixT3(R;σ)with endomorphisms and gave the necessary and sufficient conditions of which T3(R;σ)is J-quasipolar.This paper contains the following parts:The first part:The background of J-quasipolar 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 J-quasipolar rings,We opti-mized the conditions of its definition.The properties of the J-quasipolar rings are studied in more detail.Further,we studied the conditions under which J-quasipolar rings and quasipolar rings are equivalent.The fourth part:Further studied the structure theorems of 3× 3 matrix T3(R;σ)with endomorphisms and considered when T3(R;σ)is J-quasipolar.The fifth part:We introduced a new type of rings called qnilpotent rings,and then extend the known results of J-quasipolar rings to qnilpotent rings.We got many useful properties of qnilpotent rings and investigated the behavior of these properties under various ring extensions.finally,we gave the sufficient conditions of which n x n upper triangular matrix ring is qnilpotent.Finally,the dissertation puts forward the further possible research which is related to the topics in this paper.