ON COMMUTATIVITY THEOREMS FOR P. I. - RINGS WITH UNITY
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Abstract
The purpose of this paper is to show how a previous commutativity theorem for general rings can be used to prove commutativity theorems for rings with unity, and to obtain several new results via this route, e.g., if a ring with unity satisfies either $x^k[x^n, y] = [x, y^m]x^\ell$ or $x^k[x^n,y] = [x,y^m]y^\ell (m > 1)$ and if either (A) $m$ and $n$ are relatively prime or (B) $n[x,y]=0$ implies $[x,y]=0$, then $R$ is commutative.
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