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A commutativity theorem for rings. (English) Zbl 0568.16017

The aim of this short paper is to prove the following theorem: ”Let m,n be fixed non-negative integers. Suppose that R satisfies the polynomial identity: \(x^ n[x,y]-[x,y^ m]=0\), R being a ring. (i) If R is left s- unital (that is for every \(x\in R\), \(x\in Rx)\), then R is commutative except the case \(m=1\) and \(n=0\). (ii) If R is right s-unital, then R is commutative except the case \(m=1\) and \(n=0\); \(m=0\) and \(n>0.''\)
Reviewer: M.Ştefănescu

MSC:

16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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