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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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