On 3-additive mappings and commutativity in certain rings

Abstract

Let R be a ring with left identity e and suitably-restrictedadditive torsion, and Z(R) its center. LetH : R R R ! R be asymmetric 3-additive mapping, and let h be the trace ofH . In this paperwe show that (i) if for eachx 2 R,hh(x); xin = hhhh(x); xi; xi;:;x i 2 Z(R)with n 1 xed, then h is commuting on R. Moreover,h is of the formh(x) = 0x3 + 1(x)x2 + 2(x)x + 3(x) for allx 2 R;where 0 2 Z(R), 1 : R ! R is an additive commuting mapping,2 : R ! R is the commuting trace of a bi-additive mapping and themapping 3 :R ! Z(R) is the trace of a symmetric 3-additive mapping;(ii) for eachx 2 R, eitherhh(x); xin = 0 or hhh(x); xin ; xm i = 0 withn 0; m 1 xed, then h = 0 on R, wherehy;xi denotes the productyx + xy and Z(R) is the center ofR. We also present the conditions whichimplies commutativity in rings with identity as motivated by the aboveresult.