A Quadratic Differential Identity with Generalized Derivations on Multilinear Polynomials in Prime Rings

详细信息    查看全文

• 作者：Francesco Rania (1)
  Giovanni Scudo (2)
  
• 关键词：16N60 ; 16W25 ; Prime rings ; differential identities ; generalized derivations
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2014
• 出版时间：May 2014
• 年：2014
• 卷：11
• 期：2
• 页码：273-285
• 全文大小：
• 参考文献：1. Beidar K.I.: Rings with generalized identities. Moscow Univ. Math. Bull. 33, 53-8 (1978)
  2. Beidar K.I., Bresar M., Chebotar M.A.: Functional identities with r-independent coefficients. Comm. Algebra 30(12), 5725-755 (2002)
  3. Beidar, K.I., Martindale, W.S. III., Mikhalev, V.: Rings with generalized identities. In: Pure and Applied Mathematics, Dekker, New York (1996)
  4. Chuang C.L.: GPI’s having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103(3), 723-28 (1988) CrossRef
  5. Chuang C.L., Lee T.K.: Rings with annihilator conditions on multilinear polynomials. Chin. J. Math. 24(2), 177-85 (1996)
  6. De Filippis V.: Product of two generalized derivations on polynomials in prime rings. Collectanea Mathematica 61(3), 303-22 (2010) CrossRef
  7. De Filippis V., Di Vincenzo O.M., Pan C.Y.: Quadratic central differential identities on a multilinear polynomial. Comm. Algebra 36(10), 3671-681 (2008) CrossRef
  8. Di Vincenzo O.M.: On the n-th centralizer of a Lie ideal. Boll. UMI 7(3-A), 77-5 (1989)
  9. Erickson T.S., Martindale W.S. III, Osborn J.M.: Prime non-associative algebras. Pac. J. Math. 60, 49-3 (1975) CrossRef
  10. Fosner M., Vukman J.: Identities with generalized derivations in prime rings. Mediterr. J. Math. 9(4), 847-63 (2012) CrossRef
  11. Herstein, I.N.: Topics in Ring Theory. University of Chicago Press, Chicago (1969)
  12. Jacobson, N.: Structure of Rings. American Mathematical Society (Providence, RI), USA (1964)
  13. Kharchenko V.K.: Differential identities of prime rings. Algebra Log. 17, 155-68 (1978) CrossRef
  14. Lanski C.: Differential identities of prime rings, Kharchenko’s theorem and applications. Contemp. Math. 124, 111-28 (1992) CrossRef
  15. Lanski C.: Quadratic central differential identities of prime rings. Nova J. Algebra Geom. 1(2), 185-06 (1992)
  16. Lanski C., Montgomery S.: Lie structure of prime rings of characteristic 2. Pac. J. Math. 42(1), 117-35 (1972) CrossRef
  17. Lee T.K.: Derivations with invertible values on a multilinear polynomial. Proc. Am. Math. Soc. 119(4), 1077-083 (1993) CrossRef
  18. Lee T.K.: Generalized derivations of left faithful rings. Comm. Algebra 27(8), 4057-073 (1999)
  19. Lee T.K.: Derivations and centralizing mappings in prime rings. Taiwan. J. Math. 1(3), 333-42 (1997)
  20. Lee T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sinica 20(1), 27-8 (1992)
  21. Leron U.: Nil and power central polynomials in rings. Trans. Am. Math. Soc. 202, 97-03 (1975) CrossRef
  22. Martindale W.S. III: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576-84 (1969) CrossRef
  23. Posner, E.C.: Derivations in prime rings. Proc. Amer. Math. Soc. 8, 1093-100 (1957)
  24. Rowen, L.H.: Polynomial Identities in Ring Theory. Accademy Press, New York (1980)
  25. Wong T.L.: Derivations with power central values on multilinear polynomials. Algebra Colloquium 3(4), 369-78 (1996)
  26. Wong T.L.: Derivations cocentralizing multilinear polynomials. Taiwan. J. Math. 1(1), 31-7 (1997)
  
• 作者单位：Francesco Rania (1)
  Giovanni Scudo (2)
  
  1. Magna Graecia University, Catanzaro, Italy
  2. University of Messina, Messina, Italy
  
• ISSN：1660-5454

文摘

Let R be a prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C, and let \({f(x_1, \ldots, x_n)}\) be a multilinear polynomial over C, not central valued on R. Suppose that d is a derivation of R and G is a generalized derivation of R such that $$G(f(r_1, \ldots, r_n))d(f(r_1, \ldots, r_n)) + d(f(r_1, \ldots, r_n))G(f(r_1, \ldots, r_n)) = 0$$ for all \({r_1, \ldots, r_n \in R}\) . Then either d =? 0 or G =? 0, unless when d is an inner derivation of R, there exists \({\lambda \in C}\) such that G(x) =? λ x, for all \({x \in R}\) and \({f(x_1, \ldots, x_n)^2}\) is central valued on R.