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Right ideals and derivations on multilinear polynomials. (English) Zbl 1028.16016

For \(R\) an associative prime ring with center \(Z(R)\) and extended centroid \(C\), the author considers \(f(x_1,\dots,x_n)\), a multilinear polynomial over \(C\) in \(n\) noncommuting variables, \(d\) a non-zero derivation of \(R\) and \(\rho\) a non-zero right ideal of \(R\).
He proves, generalizing some results of M. N. Daif and H. E. Bell [Int. J. Math. Math. Sci. 15, No. 1, 205-206 (1992; Zbl 0746.16029)] and J. Bergen [Rend. Circ. Mat. Palermo, II. Ser. 31, 226-232 (1982; Zbl 0493.16023)] the following two results:
(i) If \((d(f(r_1,\dots,r_n))-f(r_1,\dots,r_n))^m=0\) for any \(r_1,\dots,r_n\in\rho\), then \(C\rho=eRC\) for some idempotent element \(e\in\text{Soc}(RC)\) and \(f(x_1,\dots,x_n)\) is a polynomial identity for \(eRCe\).
(ii) If \((d(f(r_1,\dots,r_n))-f(r_1,\dots,r_n))^m\in Z(R)\) for any \(r_1,\dots,r_n\in\rho\), then \(C\rho=eRC\) for some idempotent element \(e\in\text{Soc}(RC)\) and either \(f(x_1,\dots,x_n)\) is central in \(eRCe\) or \(eRCe\) satisfies the standard identity \(S_4(eRCe)=0\).

MSC:

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

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