Regular Elements Of Associative Rings And Some Commutative Conditions Of Associative 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 on rings is increasingly accurate and perfect. Preliminary results of rings have been applied in practice.Consequencely,property of rings is needed to investigate .Commuta-tivity is one of important properties on rings. Study of commutativity is beneficial to discussion of other properties on rings. At the same time, commutative rings are studied in Commutative Algebra. Therefore, study of commutativity of rings become more and more important. This paper is devoted to the study of semi-prime ring and Jacobson semi-simple ring . Using properties of zero-divisors, regular elements and subdirectly irreducible ring , we obtain some results on the commutativity of semi-prime ring and Jacobson semi-simple ring as follows 1 Let R be a semi-prime ring ,a∈R and 2a is not zero-divisor,If R satisfies one of the conditions in this paper , then R is commutative. Here we would like to point out that the above results generalized the results of Zhu Jie and Yu Xianjun. 2 Let R be a semi-prime ring .If R satisfies the conditions( α_n),then R is commutative. If ,for any elements x , y₁ ,y₂,L ,y_n∈R, there is a polynomial with integer coefficients f (t),which depends on x , y1,then [[…[ x -x² f(x),y₁],y₂],L],y_n ]∈Z(R) we call the above conditions ( α_n).This paper also discussed the other central commutator conditions which is equivalent to the former. Here we would like to point out that the above results generalized the results of Guo Huaguang. 3 Let R be a Jacobson semi-simple ring, a ∈R and 2a is not zero-divisor. If R satisfies condition [( xa )n + xnan,y]∈Z(R), for any elements x ,y in R ,which n is an fixxed integer , then R is commutative.This results enriched the commutative conditions of Jacobson semi-simple ring. 4 â‘ Let f (t 1 ,t2)= t1t2k ?1 +f2(t1,t2) be a polynomial with the strong property ( FK ) for X = {t 1}.If R satisfies [ f (x,y),y]=0 for any elements x ,y in R then â‘´when k =1, R is commutative; ⑵when R has the unity, R is commutative; ⑶when R has at least one right regular element and f 2 (t 1,t2) has no terms with degree1 such that F ( x,y)n ( x,y)= F(x,y) then R iscommutative when R has at least one right regular element. Here we would like to point out that the above results generalized the results of Fu changlin ,moreover generalized some results of other references . This results enriched the commutative conditions of any ring.