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