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Derivations with Engel conditions on multilinear polynomials. (English) Zbl 0859.16031
Let $$R$$ be a prime $$K$$-algebra, where $$K$$ is a commutative ring with 1, and let $$D$$ be a nonzero derivation of $$R$$. Set $$[x,y]_k=[[x,y]_{k-1},y]$$ where $$[x,y]_1=xy-yx$$. The authors prove that if $$f\in K\{x_1,\dots,x_n\}$$ is multilinear, and if for $$k\geq 1$$ fixed and any $$A\in R^n$$, $$[D(f(A)),f(A)]_k=0$$, then $$f(A)\in Z(R)$$, the center of $$R$$, or else $$\text{char }R=2$$ and $$R$$ embeds in $$M_2(F)$$ for some field $$F$$. This generalizes [C. Lanski, Proc. Am. Math. Soc. 118, No. 3, 731-734 (1993; Zbl 0821.16037)] which obtains the same conclusion when $$f(A)$$ is replaced by $$y\in L$$, a Lie ideal of $$R$$.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 16R50 Other kinds of identities (generalized polynomial, rational, involution) 16U80 Generalizations of commutativity (associative rings and algebras) 16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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