Let \(R\) be a ring with left identity \(e\), and let \(H\) be an additive subgroup of \(R\) containing \(e\). Let \(F\colon R^n\to R\) be an \(n\)-additive map with trace \(f\). The principal theorems, all rather technical in their statements, assert that if \(R\) has appropriate restrictions on torsion and appropriate polynomials involving \(f(x)\) and powers of \(x\) are central for all \(x\in H\), then either \(f(H)=\{0\}\) or \(f(x)x=xf(x)\) for all \(x\in H\). The results are motivated by earlier work of the reviewer and \textit{J. Lucier} [Result. Math. 36, No. 1-2, 1-8 (1999; Zbl 0938.16027)].