On the commutativity of a ring with identity. (English) Zbl 0545.16015

Tong, Jingcheng

Can. Math. Bull. 27, 456-460 (1984).

Let R be a ring with identity. R satisfies one of the following properties for all x,\(y\in R:\) (I) \(xy^ nx^ my=x^{m+1}y^{n+1}\) and \(mnm!n!x\neq 0\) except \(x=0\); (II) \(xy^ nx^ m=x^{m+1}y^ n\) and \(mm!n!x\neq 0\) except \(x=0\); (III) \(x^ my^ n=y^ nx^ m\) and \(m!n!x\neq 0\) except \(x=0\); (IV) \((x^ py^ q)^ n=x^{pn}y^{qn}\) for \(n=k\), \(k+1\) and N(p,q,k)\(x\neq 0\) except \(x=0\), where N(p,q,k) is a definite positive integer. Then R is commutative.

MSC:

16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16Rxx	Rings with polynomial identity