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An anticommutativity consequence of a ring commutativity theorem of Herstein. (English) Zbl 0619.16020

An old theorem of I. N. Herstein [Can. J. Math. 9, 583-586 (1957; Zbl 0079.054)] asserts that a ring R is commutative if for each x,y\(\in R\) there exists an integer \(n=n(x,y)>1\) for which \((xy-yx)^ n=xy-yx\). The present paper gives an analogous result on anticommutativity: specifically, that R must be anticommutative if for each x,y\(\in R\) there exists an even integer \(n=n(x,y)\geq 2\) such that \((xy+yx)^ n=xy+yx\).
Reviewer: H.E.Bell

MSC:

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

Citations:

Zbl 0079.054
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