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