A NOTE ON SKEW DERIVATIONS IN PRIME RINGS
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Let m, n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : $R{\rightarrow}R$ be a skew derivation of R and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^r-x^{m+n}D(x^r)$. We prove that if $E(x)=0$ for all $x{\in}L$, then D is a usual derivation of R or R satisfies $s_4(x_1,{\ldots},x_4)$, the standard identity of degree 4.
Abstract
Let m, n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : $R{\rightarrow}R$ be a skew derivation of R and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^r-x^{m+n}D(x^r)$. We prove that if $E(x)=0$ for all $x{\in}L$, then D is a usual derivation of R or R satisfies $s_4(x_1,{\ldots},x_4)$, the standard identity of degree 4.
참고문헌 (12)
- K. I. Beidar, "Rings with Generalized Identi- ties, Pure and Applied Math." (1996), Dekker
- C.-L. Chuang, "GPIs having coefficients in Utumi quotient rings", Proc. Amer. Math. Soc, 103(3), pp.723~728 (1988)
- C.-L. Chuang, "Differential identities with automorphisms and antiautomorphisms I", J. Algebra, 149(2), pp.371~404 (1992)
- C.-L. Chuang, "Differential identities with automorphisms and antiautomorphisms II", J. Alge- bra, 160(1), pp.130~171 (1993)
- C.-L. Chuang, "Identities with a single skew derivation", J. Algebra, 288(1), pp.59~77 (2005)
- B. Dhara, "Generalized derivations and left multipliers on Lie ideals", Aequationes Math, 81, pp.251~261 (2011)
- O. M. Di Vincenzo, "A note on k-th commutators in an associative ring", Rend. Circ. Mat. Palermo (2), 47(1), pp.106~112 (1998)
- V. K. Harcenko, "Generalized identities with automorphisms", Algebra i Logika, 14(2), pp.215~237 (1975)
- N. Jacobson, "Structure of Rings" (1964), Amer. Math. Soc.
- C. Lanski, "Differential identities, Lie ideals, and Posner's theorems", Pacific J. Math, 134(2), pp.275~297 (1988)
- W. S. Martindale III, "Prime rings satisfying a generalized polynomial identity", J. Algebra, 12, pp.576~584 (1969)