Derivations And Generalized Derivations In Prime Rings

Theory of rings as an important research branch of algebraic subjects is the base of Algebraic Geometry and Algebraic Number Theorey. Now, the theory of rings has been related to other subjects. Commutativity is one of the important properties of rings and the study of commutativity is benefical to discuss some properties of rings. Recently, the study of derivation is one of the important research topics in ring theory. The properties of derivation and the structure of this ring has a profound connection in a ring, the study of derivation in Algebra is of important significance for the study of the structure of Algebra itself. Using some properties of derivations and generalized derivations on ideals and Lie ideals, and the properties of regular elements, derivations and generalized derivations in rings are studied simply, and some new conclusions are abtained in the paper.The content of this paper divided into four parts. The main results of this paper are as follows:First, the background, the source of topics, the purpose of research and theoretical significance, the internal and external actuality and the main results of this paper are elaborated.Second, some definations and conclusions which are relative to results of this paper are given. From two aspects of 2-torsion-free and 6-torsion-free are discussed respectively, some basic conditions for commutativity of prime rings have been gained, and some known conclusions of the existing have been synthesized.Third, this dissertation studies the properties of Lie ideals in prime rings, respectively from two aspects of 2-torsion-free and 6-torsion-free are discussed, some basic conditions for U ? Z in prime rings have been gained, and some known conclusions of the existing have been synthesized, getting some theorems including a number of conclusions.Last, studying generalized derivations in prime rings, some new conclusions are obtained from two aspects of ideals and Lie ideals.