Another commutativity theorem involving certain polynomial constraints. (English) Zbl 0922.16019

Kezlan, Thomas P.

Math. Jap. 48, No. 2, 287-290 (1998).

Let \(m>1\) be an integer, and let \(R\) be a ring with 1. It is proved that \(R\) must be commutative if for each \(x,y\in R\) there exists an integer \(n=n(x,y)\geq 1\) for which \([xy-y^mx^n,x]=0\). This result is motivated by earlier results of the author [Math. Jap. 36, No. 4, 785-789 (1991; Zbl 0735.16021)] and of M. A. Quadri and M. A. Khan [Math. Jap. 33, No. 2, 275-279 (1988; Zbl 0655.16021)]. The proof, involving Herstein’s hypercenter, Chacron’s cohypercenter, and Streb’s list of factor subrings of noncommutative rings, is rather ingenious.

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:

commutativity theorems; polynomial constraints; polynomial identities; hypercenters; cohypercenters

Citations:

Zbl 0735.16021; Zbl 0655.16021