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

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.


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)