Properties Of Reversible Rings

Ring theory as an important algebraic subject is the base of Algebraic Geometry and Algebraic Number Theory. Commutativity is one of important properties of rings and the study of commutativity is beneficial to discussion of other properties on rings. The study of reversible rings is the intersection of the studies of reversible rings and other ring theories, it develops and deepens the study of commutativity.Basing on the concept of reversible ring by P. M. Cohn in 1999 and the relationships between reversible rings and other rings, such as commutative rings, reduced rings and symmetric rings by Yang Lee in 2003, the reversible rings are described by the usual methods used in module researches, the properties and decision conditions of reversible rings are studied and some new conclusions are gained.The content of this paper consists of three parts. The main results are as follows:First, the background, the purpose of research and theoretical significance, the internal and external actuality and the main results of this paper are elaborated .Second, some basic definitions and theorems which are relative to this paper are given, the proofs in follow chapter based on these definitions and conclusions. The concept of tensor product is introduced to the trivial extension by using the properties of radical , homomorphism of ring and homomorphism of modules and the properties of tensor product of reversible rings are revealed under the conditions of mode. At the same time, a class of ring of fractions are constructed by using central element and its influence on the properties of reversible rings are discussed.Finally, decision conditions of reversible rings are given by using usual tools (such as radical, nilpotent etc.) of commutativity. One decision condition is given by the characters of substructure of reversible rings, such as the character of internal ideal of reversible rings, its application scope is widespread but the actual operation is not easy to carry out. Another is given by adding external conditions, such as adding the polynomial condition, its application scope is limited but it is operated easily and more significance in practical.