Commutativity of one sided \(s\)-unital rings through a Streb's result (Q1363340)

The following theorem is proved: Let \(m,k,n\), and \(s\) be fixed nonnegative integers with \(k\) and \(n\) not both 1. If \(R\) is a left (resp. right) \(s\)-unital ring satisfying the identity \([(x^my^k)^n-x^sy,x]=0\), (resp. \([(x^my^k)^n-yx^s,x]=0\)) then \(R\) is commutative. It is also shown that \(m,k,n\), and \(s\) can be allowed to vary with \(x\) and \(y\), provided that \(R\) also has the property that for each \(x,y\in R\), there exist \(f(X),g(X)\in X^2\mathbb{Z}[X]\) for which \([x-f(x),y-g(y)]=0\); and there are some other theorems of similar character.