Substructures And Commutativity Of Associative Rings

As an important algebraic subject, Ring theory is the base of Algebraic Geometry and Algebraic Number Theory. Many other relative subjects are related to Rings. Commutativity is one of the important properties of rings. The study of commutativity is beneficial to the discussion of other properties of rings. At the same time, commutative rings are studied in Commutative Algebra in nature. Therefore, the study of commutativity of rings is very important.This article studied the commutativity of substructures with the related knowledge of zero-divisors, normal elements, subdirectly irreducible rings and density theorem and so on by using the usual tools (such as radical, nilpotent etc.) of commutativity. In particular, the commutativity of general semiperiodic rings and similar property rings are studied. Some wide commutative conditions of substructures of associative rings and general semiperiodic rings are obtained when some rings satisfy variable identities equation in this paper.The content of this paper is consisted of four parts. The main results are as follows:Firstly, the background, the purpose of research and significance, the internal and external actuality and the main results of this paper are described. Secondly, some basic definitions and conclusions which are relative to this paper are given. The commutative conditions of semiprime rings are generalized by studying the problem of commutativity of semiprime rings which satisfied binomial and trinomial. The commutative conditions of semiprime rings are studied in its left deal.Thirdly, the definition of semiperiodic rings is generalized, the commutativity of general semiperiodic rings is mainly discussed and some commutative conditions of general semiperiodic rings are gained.Finally, the commutativity of rings with F_k is limited on certain construction in this paper. The commutative conditions of rings under the condition of (2 m ? 1)-torsion-free and non- torsion-free are proved to be tenable.