Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T03:59:58.505Z Has data issue: false hasContentIssue false
  • English
  • Français

ANNIHILATORS OF POWER VALUES OF GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS

Part of: Radicals and radical properties of rings Rings and algebras with additional structure

Published online by Cambridge University Press:  19 June 2009

VINCENZO DE FILIPPIS
VINCENZO DE FILIPPIS*
Affiliation:
Dipartimento di Scienze per l’Ingegneria e per l’Architettura, Faculty of Engineering, University of Messina, 98166, Messina, Italy (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x1,…,xn) be a noncentral multilinear polynomial over C, m≥1 a fixed integer, a a fixed element of R, g a generalized derivation of R. If ag(f(r1,…,rn))m=0 for all r1,…,rn∈I, then one of the following holds:

  1. (1) aI=ag(I)=(0);

  2. (2) g(x)=qx, for some q∈U and aqI=0;

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

  4. (4) g(x)=cx+[q,x] for all x∈R, where c,q∈U such that cI=0 and [q,I]I=0.

Keywords

prime ringsdifferential identitiesgeneralized derivations

MSC classification

Secondary: 16N60: Prime and semiprime rings 16W25: Derivations, actions of Lie algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 217 - 232
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1] Argaç, N., De Filippis, V. and Inceboz, H. G., ‘Generalized derivations with power central values on multilinear polynomials on right ideals’, Rend. Sem. Mat. Univ. Padova 120 (2008), 59–71.CrossRefGoogle Scholar
[2] Bresar, M., ‘A note on derivations’, Math. J. Okayama Univ. 32 (1990), 83–88.Google Scholar
[3] Chang, C. M., ‘Power central values of derivations on multilinear polynomials’, Taiwanese J. Math. 7(2) (2003), 329–338.CrossRefGoogle Scholar
[4] Chang, C. M. and Lee, T. K., ‘Annihilators of power values in prime rings’, Comm. Algebra 26(7) (1998), 2091–2113.CrossRefGoogle Scholar
[5] Chang, C. M. and Lee, T. K., ‘Additive subgroup generated by polynomial values on right ideals’, Comm. Algebra 29(7) (2001), 2977–2984.CrossRefGoogle Scholar
[6] Chuang, C. L., ‘GPIs having coefficients in Utumi quotient rings’, Proc. Amer. Math. Soc. 103(3) (1988), 723–728.Google Scholar
[7] Chuang, C. L. and Lee, T. K., ‘Rings with annihilator conditions on multilinear polynomials’, Chinese J. Math. 24(2) (1996), 177–185.Google Scholar
[8] Faith, C. and Utumi, Y., ‘On a new proof of Litoff’s theorem’, Acta Math. Acad. Sci. Hung. 14 (1963), 369–371.CrossRefGoogle Scholar
[9] Felzenszwalb, B., ‘On a result of Levitzki’, Canad. Math. Bull. 21 (1978), 241–242.CrossRefGoogle Scholar
[10] Jacobson, N., Structure of Rings (American Mathematical Society, Providence, RI, 1964).Google Scholar
[11] Hvala, B., ‘Generalized derivations in rings’, Comm. Algebra 26(4) (1998), 1147–1166.Google Scholar
[12] Kharchenko, V. K., ‘Differential identities of prime rings’, Algebra Logic 17 (1978), 155–168.CrossRefGoogle Scholar
[13] Lee, T. K., ‘Semiprime rings with differential identities’, Bull. Inst. Math. Acad. Sinica 20(1) (1992), 27–38.Google Scholar
[14] Lee, T. K., ‘Generalized derivations of left faithful rings’, Comm. Algebra 27(8) (1999), 4057–4073.CrossRefGoogle Scholar
[15] Lee, T. K. and Lin, J. S., ‘A result on derivations’, Proc. Amer. Math. Soc. 124(6) (1996), 1687–1691.CrossRefGoogle Scholar
[16] Lee, T. K. and Shiue, W. K., ‘Identities with generalized derivations’, Comm. Algebra 29(10) (2001), 4435–4450.CrossRefGoogle Scholar
[17] Leron, U., ‘Nil and power central polynomials in rings’, Trans. Amer. Math. Soc. 202 (1975), 97–103.CrossRefGoogle Scholar
[18] Martindale III, W. S., ‘Prime rings satisfying a generalized polynomial identity’, J. Algebra 12 (1969), 576–584.CrossRefGoogle Scholar