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Two elementary commutativity theorems for generalized Boolean rings. (English) Zbl 0880.16015

Let \(R\) be an associative ring with unity satisfying the identity \([x^m-x^n,y]=0\). The author proves the commutativity of \(R\) if \(R\) satisfies one of the following conditions: (i) \(m\), \(n\) are relatively prime integers, one of which is even; (ii) \(R\) is a 2-torsion free ring and \(m=2^k(q+1)\), \(n=2^k\cdot q\).

MSC:

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)
06E20 Ring-theoretic properties of Boolean algebras
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