On Commutativity of Prime Rings with Symmetric Left θ-3- Centralizers

Let R be an associative ring with center Z(R) , I be a nonzero ideal of R and  be an automorphism  of R . An 3-additive mapping M:RxRxR R is called a symmetric left -3-centralizer if M(u1y,u2 ,u3)=M(u1,u2,u3)(y) holds for all  y, u1, u2, u3 R . In this paper , we shall investigate the  commutativity of prime rings admitting symmetric left -3-centralizer satisfying any one of the following conditions : (i)M([u ,y], u2, u3)  [(u), (y)] = 0 (ii)M((u ∘ y), u2, u3)  ((u) ∘ (y)) = 0 (iii)M(u2, u2, u3)  (u2) = 0 (iv) M(uy, u2, u3)  (uy) = 0 (v) M(uy, u2, u3)  (uy) For all u2,u3 R and u ,y I