Symmetric Bi-derivations and Commutativity of Prime Rings

Abstract:

      Let R be a ring with center Z(R) . A mapping D:R×R→R is called a symmetric bi-derivation, if D(x,y)=D(y,x),D(x+y,z)=D(x,z)+D(y,z) and D(xy , z ) = D (x , z ) y+ xD (y , z ) for all x , y , z ∈R . We show that a prime ring R with char R ≠ 2, 3, admitting a nonzero symmetric bi-derivation D , is commutative if either [ x²,D (x , x ) ] ∈ Z (R ) for all x∈R or x D (x , x ) ±D (x , x ) x ∈ Z (R ) for all x ∈R.