A commutativity theorem for division rings and an extension of a result of Faith

Abstract

It is shown that if a division ring is such that for every element a there exist co-monic integer polynomials fa, Ga of different orders so that fa(a)/Ga(a) is central then the ring is commutative. We extend this result to reduced PI rings, and show that it breaks down slightly for primitive rings, where the exceptions are characterized. This extends some earlier theorems of Herstein and Faith.