| Let R be a semi-prime ring, C be the center of R. Let Fi (x, y) (i = 1, 2) be a product of the m times x's and n times y's.In this paper following theorem is proved: (I ) implies (Ⅱ), where( Ⅰ )If f1(x,y) -f2(x,y) ∈C for every x,y in R, then R is commutative;(Ⅱ)If f1 (x,y) + f2(x, y) ∈C for every x,y in R, then R is commutative.Thus very short proves of some theorems of references[5], [8], [9] are be given. |
