On commutativity of one sided \(s\)-unital rings with some polynomial constraints. (English) Zbl 0814.16030

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 \([x- g(x), y-h(y)] = 0\).


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)