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Derivations on multilinear polynomials in semiprime rings. (English) Zbl 0943.16016

The main result of the authors concerns a 2-torsion free semiprime algebra \(R\) over a commutative ring \(K\) with \(1\), and \(f(x_1,\dots,x_n)\in K[x_1,\dots,x_n]\) a nonzero multilinear polynomial. If \(D\) is a nonzero derivation of \(R\) so that \([D(f(r_1,\dots,r_n)),f(r_1,\dots,r_n)]\) is zero or invertible in \(R\) for all \(r_j\in R\), then either \(R\) is a division ring, for all \(r_j\in R\) \(f(r_1,\dots,r_n)\) is central, or for all \(r_j\in R\) \([f(r_1,\dots,r_n),r_{n+1}]D(R)=0\). A consequence is that when \(R\) is prime and \([D(y),y]\) is zero or invertible for every \(y\in L\), a noncentral Lie ideal of \(R\), then \(R\) is a division ring.

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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[1] Beidar K.I.Rings of quotients of semprime ringsVestuik Moskov Univ. Ser. I Mat. Mekh 1978 33 36 42 English transl. in Trausl. Moscow Univ. Math. Bull. 33 (1979), 29-34
[2] Beider K.I., Extended Jacobson deusity Theorem for rings with derivations and automorphisms
[3] Beidar K.I., Pure and Applied Math (1996)
[4] DOI: 10.4153/CJM-1983-016-0 · Zbl 0522.16031 · doi:10.4153/CJM-1983-016-0
[5] Di Vincenzo O.M., Boll. UMI 3 pp 77– (1989)
[6] Herstein I.N., Topics in ring theory (1969) · Zbl 0232.16001
[7] Herstein I.N., Rings with involution (1976) · Zbl 0343.16011
[8] Jacobson N., Structure of rings 37 (1964)
[9] DOI: 10.1090/S0002-9939-1993-1132851-9 · doi:10.1090/S0002-9939-1993-1132851-9
[10] DOI: 10.1090/S0002-9939-96-03351-5 · Zbl 0859.16031 · doi:10.1090/S0002-9939-96-03351-5
[11] Lee T.K., Bull. Inst. Math. Acad. Sin 8 pp 27– (1992)
[12] DOI: 10.1090/S0002-9939-1993-1156472-7 · doi:10.1090/S0002-9939-1993-1156472-7
[13] DOI: 10.1090/S0002-9947-1975-0354764-6 · doi:10.1090/S0002-9947-1975-0354764-6
[14] Polcino Milies C., Atas de XI Eschola de Algebra pp 92– (1990)
[15] DOI: 10.1090/S0002-9939-1957-0095863-0 · doi:10.1090/S0002-9939-1957-0095863-0
[16] DOI: 10.1090/S0002-9939-1990-1007517-3 · doi:10.1090/S0002-9939-1990-1007517-3
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