TWO COMMUTATIVITY RESULTS FOR SEMI-PRIME RINGS

1.Introduction.In 1980.V.Gupta 2] provedTheorem A Let R be a semi-prime ring with unit satisfying(i)x~n,y]-x,y~n]∈Z(R) (ii)x~(n+1),y]-x,y~(n+1)]∈Z(R)for all x,y∈R and a fixed integer n>1,then R is commutative.This theoremimproved a theorem which was established by Harmanci 4] in 1977 that if aring R with unit satisfies the identity(i)x~n,y]=x,y~n] (ii)x~(n+1),y]=x,y~(n+1)] for all x,y∈Rand a fixed integer n>1,then R is commutative.Later,in 1982,Guo Yuan