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Posner’s second theorem, multilinear polynomials and vanishing derivations. (English) Zbl 1059.16024
Let \(K\) be a commutative ring with \(1\) and \(f(X)\in K\{X\}\) a multilinear polynomial in \(n\) noncommuting indeterminates \(X\) over \(K\). Let \(R\) be a prime \(K\)-algebra with \(\text{char\,}R\neq 2\) and for \(\alpha\in R^n\) let \(f(\alpha)\) be the evaluation of \(f(X)\) at the entries of \(\alpha\). The authors prove that if for nonzero derivations \(D\) and \(E\) of \(R\) and all \(\alpha\in R^n\), \(E(D(f(\alpha))\alpha-\alpha D(f(\alpha)))=0\) then all \(f(\alpha)\) are in the center of \(R\).

MSC:
16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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