On commutativity of one sided \(s\)-unital rings with some polynomial constraints (Q1338188)

Let \(m\), \(n\) and \(p\) be fixed nonnegative integers. Call the ring \(R\) a \((*)\)-ring (resp. a \((*)'\)-ring) if for each \(x, y \in R\) there exists \(f(t) \in t^ 2 \mathbb{Z} [t]\) such that \([x^ my - x^ nf(x^ my)x^ p,x] = 0\) (resp. \([yx^ m - x^ nf(x^ my)x^ p,x] = 0\)). It is proved that left \(s\)-unital \((*)\)-rings and right \(s\)-unital \((*)'\)-rings must be commutative. This theorem can be extended to the case where \(m\), \(n\) and \(p\) vary with \(x\) and \(y\), provided we impose an additional hypothesis due to Chacron -- namely for each \(x, y \in R\) there exist \(g(t)\), \(h(t) \in t^ 2\mathbb{Z}[t]\) for which \([x- g(x), y-h(y)] = 0\).