Commutative Conditions Of Rings

As an important algebraic subject, rings are the base on Algebraic Geometry and Algebraic Number Theory. Rings are concerned about many other subjects. With development of science and technology, theory in rings is increasingly accurate and perfect and preliminary results of rings have been applied in practice. Commutativity is one of important properties in rings. Study of commutativity is beneficial to discussion of other properties on rings. Property of rings is needed to investigate. At the same time, commutative rings are studied in Commutative Algebra. Therefore, study of commutativity of rings becomes more and more important.We get some new results on commutativity of rings by studying the Koethe semi-simple ring, semi-prime ring and arbitrary ring. Using properties of zero-divisors, regular elements, center and subdirectly irreducible ring, We get some further conclusions on aspects of particular rings and gain some new conclusions. This paper consists of three parts. The main contents in this paper are as follows:1. I show the background, significance of the theories, the internal, external actuality and describe the main results of this paper. It is ready for the following to drawing several conclusions.2. first, I give some conceptions and conclusions which are relative to this paper. Second, I put forward and strictly proved one commut-ative conditions of arbitrary ring are given by studying the Herstein theorem and get a further conclusion based on it.3. I study commutative conditions of rings with variable identical equation. It consists of two conclusions. In this chapter, first, I talked about commutative conditions of rings with central regular elements and I further extend commutative conditions of Herstein theory and then I further talked about other commutative conditions of rings with central regular elements.