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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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