A commutativity theorem for associative rings. (English) Zbl 0839.16030

Ashraf, Mohammad

Arch. Math., Brno 31, No. 3, 201-204 (1995).

Let \(m>1\) and \(s\geq 1\) be fixed positive integers, and \(R\) be an associative ring with unity such that for any \(x,y\in R\), \(m[x,y]=0\) implies \([x,y]=0\). The author proves that a ring \(R\) is commutative if \(R\) satisfies one of the following conditions: 1) for each \(x\in R\) there exist nonnegative integers \(p\), \(q\), \(n\), \(r\) such that for all \(y\in R\), \(x^p[x^n,y]x^q=y^s[x,y^m]x^r\); 2) for each \(x\in R\) there exist nonnegative integers \(p\), \(q\), \(n\), \(r\) such that for all \(y\in R\) \(x^p[x^n,y]x^q=x^r[x,y^m]y^s\).

Reviewer: Yu.N.Mal’tsev (Barnaul)

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:

commutativity theorem; commutator constraints