A commutativity theorem involving certain polynomial constraints. (English) Zbl 0735.16021

Kezlan, Thomas P.

Math. Jap. 36, No. 4, 785-789 (1991).

It is shown that if \(n\) is a positive integer and \(R\) is an associative ring with unity, having the property: (*) given \(x,y\) in \(R\) there exists an integer \(m=m(x,y)>1\) such that \([xy-y^ mx^ n,x]=0\) then \(R\) must be commutative. If we consider \(m\) independent of \(x\) and \(y\), we obtain a theorem which has been proved by M. A. Quadri and M. Khan [Math. Jap. 33, 275-279 (1988; Zbl 0655.16021)]. The case in which both \(m\) and \(n\) are allowed to depend on \(x\) and \(y\) remains open.

Reviewer: M.Guţan

MSC:

16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)

Keywords:

commutativity; polynomial constraints

Citations:

Zbl 0655.16021