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Power central values of derivations on multilinear polynomials. (English) Zbl 1058.16032
Author’s summary: Let $$R$$ be a prime ring with extended centroid $$C$$, $$\rho$$ a nonzero right ideal of $$R$$, $$d$$ a nonzero derivation of $$R$$, $$f(X_1,\dots,X_t)$$ a multilinear polynomial over $$C$$, $$a\in R$$ and $$n$$ a fixed positive integer. (I) If $$ad(f(x_1,\dots,x_t))^n=0$$ ($$d(f(x_1,\dots,x_t))^na=0$$) for all $$x_1,\dots,x_t\in\rho$$, then either $$a\rho=0$$ ($$a=0$$ resp.), $$d(\rho)\rho=0$$ or $$\rho C=eRC$$ for some idempotent $$e$$ in the socle of $$RC$$ such that $$f(X_1,\dots,X_t)$$ is central-valued on $$eRCe$$. (II) If $$ad(f(x_1,\dots,x_t))^n\in C$$ ($$d(f(x_1,\dots,x_t))^na\in C$$) for all $$x_1,\dots,x_t\in\rho$$ and $$ad(f(y_1,\dots,y_t))^n\neq 0$$ ($$d(f(y_1,\dots,y_t))^na\neq 0$$) for some $$y_1,\dots,y_t\in\rho$$, then either $$f(\rho)\rho=0$$ or $$f(X_1,\dots,X_t)$$ is central-valued on $$RC$$ unless $$\dim_CRC=4$$.

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