Certain polynomial identities and commutativity of rings. (English) Zbl 0958.16035

Let \(R\) be a ring. Let us denote by \((P)\) the following property: For all \(x,y\in R\), let \([x,y]=y^s[x^n,y^m]^ky^t\), where \(m>1\), \(k>0\), \(s\geqq 0\), \(t\geqq 0\) are fixed non-negative integers. The author shows that a ring \(R\) is commutative if and only if \(R\) satisfies the property \((P)\). Further properties making \(R\) commutative are introduced here.


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16U80 Generalizations of commutativity (associative rings and algebras)
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