On a class of infinite rings (Q2731592)

Let \(f=f(x_1,x_2,\dots,x_n)\) be a polynomial in \(n\) noncommuting indeterminates with integer coefficients. The ring \(R\) is called an \(f^\#\)-ring if for every \(n\) infinite subsets \(X_1,X_2,\dots,X_n\) of \(R\), there exist \(r_i\in X_i\) for which \(f(r_1,r_2,\dots,r_n)=0\); \(R\) is called an \(f\)-ring if \(f\) is a polynomial identity for \(R\). The authors prove that for \(f(x_1,x_2,\dots,x_n)=x^{k_1}_1x^{k_2}_2\cdots x^{k_n}_n\), every infinite \(f^\#\)-ring is an \(f\)-ring. This result has several analogues in group theory, proved by the authors among others; and it was prompted by a result of the reviewer, \textit{A. A. Klein} and \textit{L.-C. Kappe}, stating that for \(g(x_1,x_2)=x_1x_2-x_2x_1\), every infinite \(g^\#\)-ring is a \(g\)-ring [Acta Math. Hung. 77, No. 1-2, 57-67 (1997; Zbl 0905.16011)].