ABSTRACT. The following theorem is proved: Let r r(y) > 1, s, and be non-negative integers. If R is a left s-unital ring satisfies the polynomial identity [xy x'y"x t, x] 0 for every x, y E R, then R is commutative. The commutativity of a right s-unital ring satisfying the polynomial identity [x/-yrxt, X] 0 for all x, y E R, is also proved.

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We discuss the commutativity of certain rings with unity 1 and one-sided s-unital rings under each of the following conditions: xr [xs, y] = ±[x, yt ]xn , xr [xs, y] = ±xn [x, yt ], xr [xs, y] = ±[x, yt ]ym , and xr [xs, y] = ±ym [x, yt ], where r, n, and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s[x, y] = 0 implies [x, y] = 0. Further, we improve the result of [Jacobson, Structure of rings, Colloq. Publ., 1964, Theorem 3] and reprove several recent results.