Commutativity On Semi-prime Ring

As an important algebraic subject,rings are the base of Algebraic Geometry and Algebraic Theory.Rings are concerned about many other subjects . With development of science and technology, theory of rings is increasingly accurate and perfect. Preliminary results of rings have been applied in practice. Consequencely, property of rings need to be investigated. Commutativity is one of important properties of rings.Study of commutativity is beneficial to discussion of other properties of rings. At the same time,commutative rings arestudied in Commutative Algebra.Therefore,study of the commutativity of rings become more and more important.The main results in this paper as follow: 1 Let R is semiprime ring with characteristi2 and Z (R) be the center of R, if R satisfies one of the following condtions, then R is commutative.(1) a∈0, for all x∈R, a2x2 + ax2a ∈ Z(R);(2) a∈R,a2 0, for all xeR, x2a + xa2x∈Z{R).2 Let R is semiprime ring, a∈R and 2a is not zero divisor , if R satisfies one of the following conditions, then R is commutative.1) for all x∈R, (xa)2+x2a2 ∈Z(R);2) for all x∈R, (xa)2+a2x2 ∈Z(R);3) for all x∈R, (xa)2 + xa2x2Z(R);4) for all x∈R, (xa)2 +ax2a ∈ Z(R).