Let \(R\) be a prime ring with extended centroid \(C\), \(T\) a nonzero right ideal of \(R\), and \(f\in C\{X_1,\dots,X_t\}\) with constant term zero so that \(f\) is not central valued on \(RC\). Let \(f(T)\) denote the additive subgroup of \(RC\) generated by all evaluations of \(f\) using elements from \(T\). The author proves that if \(\dim_Cf(T)C\) is finite, then \(\dim_CRC\) is finite, and if \(f(T)\neq 0\) is finite, then \(R\) is finite. In particular if \([T,T]\) is finite, or more generally if \(f=[x,y]_m=[[x,y]_{m-1},y]\) and \(f(T)\) is finite, then \(R\) is commutative or finite.