A commutativity theorem for associative rings. (English) Zbl 0655.16021

Quadri, Murtaza A.; Khan, Moharram A.

Math. Jap. 33, No. 2, 275-279 (1988).

The reviewer has proved that a ring \(R\) must be commutative if for each \(x,y\in R\) there exist integers \(m=m(x,y)\geq 1\) and \(n=n(x,y)\geq 1\) such that \(xy=y^mx^n\) [Can. J. Math. 28, 986–991 (1976; Zbl 0319.16028)]. The present authors extend this result as follows: if \(R\) has 1 and there exist fixed integers \(m>1\) and \(n\geq 1\) such that \([xy-y^mx^n, x]=0\) for all \(x,y\in R\), then \(R\) is commutative. The hypothesis that \(R\) has 1 cannot be deleted from their theorem.

Reviewer: H.E.Bell

MSC:

16U70

Center, normalizer (invariant elements) (associative rings and algebras)

Keywords:

commutativity

Citations:

Zbl 0336.16033; Zbl 0319.16028