Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero


Covered by:
Web of Science - Emerging Sources Citation Index
Scopus
MathSciNet


CiteScore 2017: 0.28
SCImago Journal Rank (SJR) 2017: 0.231
Source Normalized Impact per Paper (SNIP) 2017: 0.443
Mathematical Citation Quotient (MCQ) 2017: 0.12
ICV 2017: 121.78



Open Access
Online
ISSN
2391-4661
See all formats and pricing
More options …
Volume 47, Issue 1

Issues

Cocentralizing Generalized Derivations On Multilinear Polynomial On Right Ideals Of Prime Rings

Vincenzo De Filippis / Basudeb Dhara
Published Online: 2014-03-07 | DOI: https://doi.org/10.2478/dema-2014-0002

Abstract

Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R ƒ (x1… xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(ƒ (r)) ƒ (r)- ƒ (r)H(ƒ (r)) ∈ C, for all r =(r1,….,rn) ∈ In. Then one of the following holds:

1. there exist a; b; p ∈ U and α C such that G(x)= ax + [p, x] and H(x) = bx, for all x ∈ R, and (a-b)I=(0)=(a + p- α)I;

2. R satisfies s4, the standard identity of degree 4, and there exist a; a' ∈ U, α,β ∈ C such that G(x) =ax + xa' + αx and H(x) = a'x - xa +βx, for all x ∈ R;

3. R satisfies s4 and there exist a; a' ∈ U, and d : R → R, a derivation of R, such that G(x) = ax + d(x) and H(x)= xa'- d(x), for all x ∈ R, with a + a' ∈ C;

4. R satisfies s4 and there exist a; a' ∈ U, and d : R → R, a derivation of R, such that G(x) = xa + d(x) and H(x) = ax' - d(x), for all x ∈ R, with a - a' ∈ C;

5. there exists e2= e ∈ Soc(RC) such that I = eR and one of the following holds:

(a) [ƒ (x1 ,…., xn); xn + 1] xn+2 is an identity for I;

(b) char (R) = 2 and s4(x1; x2; x3; x4)x5 is an identity for I;

(c) [ƒ (x1 , …, xn)2; xn+1]xn+2 is an identity for I and there exist a, a', b, b' ∈ U,α ∈ C and d : R → R, a derivation of R, such that G(x) = ax + xa' + d(x), H(x)=bx + xb' - d(x), for all x ∈ R, with (a - b' - α) I=(0)=( b-a'-α )I

Keywords : prime rings; differential identities; generalized derivations

References

  • [1] A. Argac, L. Carini, V. De Filippis, An Engel condition with generalized derivations on Lie ideals, Taiwanese J. Math. 12(2) (2008), 419-433.Google Scholar

  • [2] A. Argac, V. De Filippis, Actions of Generalized Derivations on Multilinear Polynomials in Prime Rings, Algebra Colloquium 18 (spec. 1) 2011, 955-964.CrossrefGoogle Scholar

  • [3] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385-394.Google Scholar

  • [4] C. M. Chang, T. K. Lee, Annihilators of power values of derivations in prime rings, Comm. Algebra 26(7) (1998), 2091-2113.CrossrefWeb of ScienceGoogle Scholar

  • [5] C. L. Chuang, The additive subgroup generated by a polynomial, Israel J. Math. 59(1) (1987), 98-106.Google Scholar

  • [6] C. L. Chuang, GPI’s having coefficients in Utumi quotient rings, Proc. Amer. Math.Soc. 103(3) (1988), 723-728.Google Scholar

  • [7] C. L. Chuang, T. K. Lee, Rings with annihilator conditions on multilinear polynomials, Chinese J. Math. 24(2) (1996), 177-185.Google Scholar

  • [8] B. Dhara, V. De Filippis, Notes on generalized derivations on Lie ideals in prime rings, Bull. Korean Math. Soc. 46(3) (2009), 599-605.Google Scholar

  • [9] O. M. Di Vincenzo, On the n-th centralizer of a Lie ideal, Boll. Un. Mat. Ital. A (7) 3 (1989), 77-85.Google Scholar

  • [10] T. S. Erickson, W. S. Martindale III, J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), 49-63.Google Scholar

  • [11] C. Faith, Y. Utumi, On a new proof of Litoff’s theorem, Acta Math. Acad. Sci. Hung. 14 (1963), 369-371.CrossrefGoogle Scholar

  • [12] I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago, IL, 1969.Google Scholar

  • [13] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Pub., 37, Amer. Math.Soc., Providence, RI, 1964.Google Scholar

  • [14] V. K. Kharchenko, Differential identity of prime rings, Algebra Logic 17 (1978), 155-168.Google Scholar

  • [15] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118(3) (1993), 731-734.CrossrefGoogle Scholar

  • [16] C. Lanski, S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J.Math. 42(1) (1972), 117-135.Google Scholar

  • [17] T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20(1) (1992), 27-38.Google Scholar

  • [18] T. K. Lee, Left annihilators characterized by GPIs, Trans. Amer. Math. Soc. 347 (1995), 3159-3165.Google Scholar

  • [19] T. K. Lee, Power reduction property for generalized identities of one-sided ideals, Algebra Colloq. 3 (1996), 19-24.Google Scholar

  • [20] T. K. Lee, Derivations with Engel conditions on polynomials, Algebra Colloq. 5(1) (1998), 13-24.Web of ScienceGoogle Scholar

  • [21] T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27(8) (1999), 4057-4073.Google Scholar

  • [22] T. K. Lee, W. K. Shiue, Derivations cocentralizing polynomials, Taiwanese J. Math. 2(4) (1998), 457-467.Google Scholar

  • [23] P. H. Lee, T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad.Sinica 23 (1995), 1-5.Google Scholar

  • [24] U. Leron, Nil and power central valued polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103.Google Scholar

  • [25] W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J.Algebra 12 (1969), 576-584.CrossrefGoogle Scholar

  • [26] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.Google Scholar

  • [27] L. M. Rowen, Polynomial Identities in Ring Theory, Pure and Applied Mathematics 84, Academic Press, New York, 1980.Google Scholar

  • [28] T. L. Wong, Derivations with cocentralizing multilinear polynomials, Taiwanese J.Math. 1 (1997), 31-37. Google Scholar

About the article

Published Online: 2014-03-07

Published in Print: 2014-03-01


Citation Information: Demonstratio Mathematica, Volume 47, Issue 1, Pages 22–36, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.2478/dema-2014-0002.

Export Citation

© 2015 by Walter de Gruyter Berlin/Boston. This content is open access.

Comments (0)

Please log in or register to comment.
Log in