Nil polynomials of prime rings (Q1355547)

The authors prove a neat and decisive result about polynomials with nilpotent values. Let \(R\) be a prime ring with extended centroid \(C\), and having no nonzero nil right ideal. If for \(f(x_1,x_2,\dots,x_m)\in C\{X\}\), the free algebra over \(C\) in noncommuting indeterminates, \(f(r_1,r_2,\dots,r_m)^n=0\) for all \(r_i\in R\) and \(n=n(r_1,\dots,r_m)\), then either \(f\) is an identity for \(R\), that is \(f(R^m)=0\), or \(R\cong M_k(\text{GF}(p^t))\).