Some Commutative Theorems Of Ring

This paper is devoted to the study of K(O|..)the semi-simple ring and semi-prime ring. Using properties of zero-divisors, regular elements and diaphysis elements , we obtain some results on the commutativity of K(O|..)the semi-simple ring and semi-prime ring as follows:Let R be aK(O|..)the semi-simple ring satisfying condition (A), then R is commutative.Let R be a semi-prime ring. If R satisfies condition (A) and the coefficients of the f(t₁,t₂) satisfy one of the following conditions:(1) a_xb and a_yb are relatively prime numbers;(2) a_xe and a_ye are relatively prime numbers;and n is a fixed positive integer, then R is commutative. Here we would like to point out that the above two results generalize the results of Kezlan, Dai Yuejing, and Liu Zheyi et al.3 If a semi-prime ring R satisfies [f^x(x,y)(x,y)±g(x,y),z]∈C for any z, then R is commutative.Most of the relevant results of Awtar, Quadri, Guo Yuanchun and Jiu Qizhang can be induced from result 3. In this paper, we make the assumption that the polynomial f(t₁,t₂) can be denoted by the sum of two polynomials f₁(t₁,t₂) and f₂(t₁,t₂). Here, f₁(t₁,t₂)is the minimum degree part of f, which for t₁ is of degree k₁ and for t₂ is of degree k₂(k₁+k₂=n), and the sum of the coefficients each term is 1 (or -1), and the minimum degree of f₂(t₁,t₂) is larger than n, and the degree of t₁ in each term of f₂(t₁,t₂) is the p times of k₁, where p is a positive integer larger than 1.We say that a ring R satisfies condition (A) is that:For any a,b,c∈R,there exists the polynomial depending on a,b,c such that[f(a,b),c]=0where k₁ and k₂=n-k₁ in f may depend on a,b,c.