Rings with a setwise polynomial-like condition

Abstract

Let R be an infinite ring. Here, we prove that if 0R be- longs to {x1x2 · · · xn | x1,x2,...,xn ε X} for every infinite subset X of R, then R satisfies the polynomial identity xn = 0. Also, we prove that if 0r belongs to {x1x2 · · · xn - xn+1 | x1, x2,..., xn, xn+1 ε X} for every infinite subset X of R, then xn = x, for all x ε R. © 2012 Iranian Mathematical Society.