Di Vincenzo, O. M. Derivations and multilinear polynomials. (English) Zbl 0738.16016 Rend. Semin. Mat. Univ. Padova 81, 209-219 (1989). 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 \(S_ 4\). Reviewer: C.Lanski (Los Angeles) Cited in 1 Document 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) Keywords:prime ring; \(f\)-radical; derivation; multilinear and homogeneous polynomial; centroid; standard polynomial identity; center; Lie ideals PDFBibTeX XMLCite \textit{O. M. Di Vincenzo}, Rend. Semin. Mat. Univ. Padova 81, 209--219 (1989; Zbl 0738.16016) Full Text: Numdam EuDML References: [1] J. Bergen - A. GIAMBRUNO, f-radical extensions of rings , Rend. Sem. Mat. Univ. Padova , 77 ( 1987 ), pp. 125 - 133 . Numdam | MR 904615 | Zbl 0623.16008 · Zbl 0623.16008 [2] J. Bergen - I.N. Herstein - J.W. Kerr , Lie ideals and derivations of prime rings , J. Algebra , 71 ( 1981 ), pp. 259 - 267 . MR 627439 | Zbl 0463.16023 · Zbl 0463.16023 [3] L. Carini , Centralizers and Lie ideals , Rend. Sem. Mat. Univ. Padova , 78 ( 1987 ), pp. 255 - 259 . Numdam | MR 934516 | Zbl 0637.16021 · Zbl 0637.16021 [4] B. Felzenswalb - A. Giambruno , Centralizers and multilinear polynomials in non-commutative rings , J. London Math. Soc. , 19 ( 1979 ), pp. 417 - 428 . MR 540054 | Zbl 0405.16011 · Zbl 0405.16011 [5] B. Felzenswalb - A. Giambruno , Periodic and nil potynomials in rings , Canad. Math. Bull. , 23 ( 1980 ), pp. 473 - 476 . MR 602605 | Zbl 0462.16007 · Zbl 0462.16007 [6] B. Felzenswalb - A. Giambruno , A commutativity theorem for rings with derivations , Pacific J. Math. , 102 ( 1982 ), pp. 41 - 45 . Article | MR 682042 | Zbl 0501.16032 · Zbl 0501.16032 [7] A. Giambruno , Rings f-radicals over P.I. subrings , Rend. Mat. , ( 1 ), 13 , VI ( 1980 ), pp. 105 - 113 . MR 590736 | Zbl 0452.16010 · Zbl 0452.16010 [8] I.N. Herstein , Rings with Involution , Univ. Chicago Press , Chicago , 1976 . MR 442017 | Zbl 0343.16011 · Zbl 0343.16011 [9] I.N. Herstein , A theorem on invariant subrings , J. Algebra , 83 ( 1983 ), pp. 26 - 32 . MR 710584 | Zbl 0514.16001 · Zbl 0514.16001 [10] 1 N. Herstein - C. PROCESI - M. SCHACHER, Algebraic valued functions on non-commutative rings , J. Algebra , 36 ( 1975 ), pp. 128 - 150 . MR 374185 | Zbl 0311.16017 · Zbl 0311.16017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.