A commutativity theorem for power-associative rings

Abstract

Let R be a power-associative ring with identity and let I be an ideal of R such that R/I is a finite field and x H y (mod I) implies x 2 = y 2 or both x and y commute with all elements of I . It is proven that R must then be commutative.Examples are given to show that R need not be commutative if various parts of the hypothesis are dropped or if "x 2 = t/ 2 " is k k replaced by "x = y " for any integer k > 2 .