Let R be a ring with left identity e and suitably-restricted additive torsion, and Z(R) its center. Let H : $R{ imes}R{ imes}R{ ightarrow}R$ be a symmetric 3-additive mapping, and let h be the trace of H. In this paper we show that (i) if for each $x{in}R$, $$n=<,;x>,;cdots,x>{in}Z(R)$$ with $ngeq1$ fixed, then h is commuting on R. Moreover, h is of the form $$h(x)=lambda_0x^3+lambda_1(x)x^2+lambda_2(x)x+lambda_3(x);for;all;x{in}R$$, where $lambda_0;{in};Z(R)$, $lambda_1;:;R{ ightarrow}R$ is an additive commuting mapping, $lambda_2;:;R{ ightarrow}R$ is the commuting trace of a bi-additive mapping and the mapping $lambda_3;:;R{ ightarrow}Z(R)$ is the trace of a symmetric 3-additive mapping; (ii) for each $x{in}R$, either $n=0;or;<n,;x^m>=0$ with $ngeq0,;mgeq1$ fixed, then h = 0 on R, where <y, x> denotes the product yx+xy and Z(R) is the center of R. We also present the conditions which implies commutativity in rings with identity as motivated by the above result.