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

Ashraf, Mohammad

Indian J. Pure Appl. Math. 25, No. 9, 963-967 (1994).

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

Reviewer: H.E.Bell (St.Catherines)

MSC:

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)

Keywords:

commutator constraints; \(s\)-unital \((*)\)-rings; right \(s\)-unital \((*)'\)- rings