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Derivations and multilinear polynomials. (English) Zbl 0738.16016

The author studies the structure of a prime ring \(R\) which is \(f\)-radical over the ring of constants of a derivation of \(R\). The main result is: Theorem 1. Let \(R\) be a prime ring with \(\hbox{char} R\neq 2\), \(D\) a nonzero derivation of \(R\), and \(f(x_ 1,\dots,x_ n)\) a multilinear and homogeneous polynomial over the centroid of \(R\). If \(R\) contains no nonzero nil right ideal, if for any \(r_ 1,\dots,r_ n\in R\) there is a positive integer \(m=m(r_ 1,\dots,r_ n)\) so that \(D(f(r_ 1,\dots,r_ n)^ m)=0\), and if when \(\hbox{char} R=p>0\) \(f(x_ 1,\dots,x_ n)\) is not an identity for \(M_ p(F)\) where \(F\) is a field and \(\hbox{char} F=p\), then \(R\) satisfies the standard polynomial identity \(S_{n+2}\), and also for all \(r_ 1,\dots,r_ n\in R\), \(f(r_ 1,\dots,r_ n)^ k\in Z\), the center of \(R\), for \(k=k(r_ 1,\dots,r_ n)\). One easy consequence of the main theorem is a nice result for Lie ideals. If \(U\) is a noncentral Lie ideal of \(R\), and if for each \(u\in U\) there is \(m=m(u)\geq 1\) so that \(D(u^ m)=0\), then \(R\) satisfies \(S_ 4\).

MSC:

16W25 Derivations, actions of Lie algebras
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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References:

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