RINGS WITH A SETWISE POLYNOMIAL-LIKE CONDITION

Let R be an infinite ring. Here, we prove that if 0R belongs to {x1 x2 · · · xn | x1, x2, . . . , xn element of X} for every infinite subset X of R, then R satisfies the polynomial identity x^n = 0. Also, we prove that if 0R belongs to {x1 x2 · · · xn−xn+1 | x1, x2, . . . , xn, xn+1 element of X} for every infinite subset X of R, then x^n = x, for all x element of R.