Commutativity Of Associative Rings With F_k-Property And Similar Property

Algebra as a fundamental pillar of mathematics, it is an important source ofmathematical thought and mathematical methods. Ring is one of the four basicstructures of algebra, the theory of ring is applied to many relevant subjects.Commutativity of ring is an important subject to study of the properties of rings, thestudy of commutativity of ring can promote study of other properties of ring, andenrich knowledge systems of ring theory, promote the development of whole of thealgebra.Using usual tools of structures of Rings (such as radical, nilopotent etc) andaccossiative knowledge of normal elements, zero-divisors, the center, subdirectlyirreducible rings and density theorem and so on, study deeply the commutativity ofring satisfies some commutator equations, in particular study the commutativity ofring withF_k property and similar property. This is the main purpose of thisarticle.This paper is divided into three parts. The main work as follows:Firstly, the background, purpose, meaning, domestic and foreign researchactuality of research subject and the main results of this paper are introduced.Secondly, some basic definitions and fundamental lemmas which are relative tothis paper are given, and we have got the similar property commutatorequation conditions which can make commutate of semi-prime ring.The last part is the focus of this paper. Commutativity of ring withproperty is further studied. Firstly the more general commutator equationconditions, which can make rings commutative, are obtained. Then, it is proved thatrings are also commutative when some substructures of this ring satisfying the sameconditions...