m-potent commutators involving skew derivations and multilinear polynomials

  • 1 Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India
  • 2 Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India
  • 3 Faculty of Science & Arts-Rabigh, King Abdulaziz University, Jeddah, Saudi Arabia
Mohammad AshrafORCID iD: https://orcid.org/0000-0001-6976-3629, Sajad Ahmad ParyORCID iD: https://orcid.org/0000-0003-3542-0219 and Mohd Arif RazaORCID iD: https://orcid.org/0000-0001-6799-8969

Abstract

Let be a prime ring, 𝒬r the right Martindale quotient ring of and 𝒞 the extended centroid of . In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,

([δ(f(x1,,xn)),f(x1,,xn)])m=[δ(f(x1,,xn)),f(x1,,xn)],

where 1<m+, f(x1,x2,,xn) is a non-central multilinear polynomial over 𝒞 and δ is a skew derivation of .

  • [1]

    L. Carini and V. De Filippis, Commutators with power central values on a Lie ideal, Pacific J. Math. 193 (2000), no. 2, 269–278.

    • Crossref
    • Export Citation
  • [2]

    L. Carini, V. De Filippis and G. Scudo, Power-commuting generalized skew derivations in prime rings, Mediterr. J. Math. 13 (2016), no. 1, 53–64.

    • Crossref
    • Export Citation
  • [3]

    C.-L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723–728.

    • Crossref
    • Export Citation
  • [4]

    C. L. Chuang, Differential identities with automorphisms and antiautomorphisms. II, J. Algebra 160 (1993), no. 1, 130–171.

    • Crossref
    • Export Citation
  • [5]

    C.-L. Chuang and T.-K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005), no. 1, 59–77.

    • Crossref
    • Export Citation
  • [6]

    V. De Filippis, A product of two generalized derivations on polynomials in prime rings, Collect. Math. 61 (2010), no. 3, 303–322.

    • Crossref
    • Export Citation
  • [7]

    V. De Filippis, M. Arif Raza and N. U. Rehman, Commutators with idempotent values on multilinear polynomials in prime rings, Proc. Indian Acad. Sci. Math. Sci. 127 (2017), no. 1, 91–98.

    • Crossref
    • Export Citation
  • [8]

    V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40 (2012), no. 6, 1918–1932.

    • Crossref
    • Export Citation
  • [9]

    V. De Filippis and S. Huang, Power-commuting skew derivations on Lie ideals, Monatsh. Math. 177 (2015), no. 3, 363–372.

    • Crossref
    • Export Citation
  • [10]

    V. De Filippis and G. Scudo, Strong commutativity and Engel condition preserving maps in prime and semiprime rings, Linear Multilinear Algebra 61 (2013), no. 7, 917–938.

    • Crossref
    • Export Citation
  • [11]

    V. De Filippis and F. Wei, Posner’s second theorem for skew derivations on multilinear polynomials on left ideals, Houston J. Math. 38 (2012), no. 2, 373–395.

  • [12]

    C. Demir and N. Argaç, Prime rings with generalized derivations on right ideals, Algebra Colloq. 18 (2011), no. Special Issue 1, 987–998.

    • Crossref
    • Export Citation
  • [13]

    N. J. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955), 19–22.

  • [14]

    B. Felzenszwalb, On a result of Levitzki, Canad. Math. Bull. 21 (1978), no. 2, 241–242.

    • Crossref
    • Export Citation
  • [15]

    V. K. Harčenko, Generalized identities with automorphisms, Algebra i Logika 14 (1975), no. 2, 215–237, 241.

  • [16]

    I. N. Herstein, A generalization of a theorem of Jacobson, Amer. J. Math. 73 (1951), 756–762.

    • Crossref
    • Export Citation
  • [17]

    I. N. Herstein, A generalization of a theorem of Jacobson. III, Amer. J. Math. 75 (1953), 105–111.

    • Crossref
    • Export Citation
  • [18]

    I. N. Herstein, The structure of a certain class of rings, Amer. J. Math. 75 (1953), 864–871.

    • Crossref
    • Export Citation
  • [19]

    I. N. Herstein, A condition for the commutativity of rings, Canadian J. Math. 9 (1957), 583–586.

    • Crossref
    • Export Citation
  • [20]

    S. Huang, Derivations with power values on multilinear polynomials, Iran. J. Sci. Technol. Trans. A Sci. 39 (2015), no. 4, 521–525.

  • [21]

    N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math. (2) 46 (1945), 695–707.

    • Crossref
    • Export Citation
  • [22]

    N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ. 37, American Mathematical Society, Providence, 1964.

  • [23]

    L. Karini and V. de Filippis, Centralizers of generalized derivations on multilinear polynomials in prime rings (in Russian), Sibirsk. Mat. Zh. 53 (2012), no. 6, 1310–1320; translation in Sib. Math. J. 53 (2012), no. 6, 1051–1060.

  • [24]

    C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 731–734.

    • Crossref
    • Export Citation
  • [25]

    T. K. Lee, Derivations with invertible values on a multilinear polynomial, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1077–1083.

    • Crossref
    • Export Citation
  • [26]

    U. Leron, Nil and power-central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97–103.

    • Crossref
    • Export Citation
  • [27]

    J. H. Maclagan-Wedderburn, A theorem on finite algebras, Trans. Amer. Math. Soc. 6 (1905), no. 3, 349–352.

    • Crossref
    • Export Citation
  • [28]

    W. S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584.

    • Crossref
    • Export Citation
  • [29]

    J. H. Mayne, Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull. 35 (1992), no. 4, 510–514.

    • Crossref
    • Export Citation
  • [30]

    E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.

    • Crossref
    • Export Citation
  • [31]

    N. Rehman and M. Arif Raza, On m-commuting mappings with skew derivations in prime rings (in Russian), Algebra i Analiz 27 (2015), no. 4, 74–86; translation in St. Petersburg Math. J. 27 (2016), no. 4, 641–650.

  • [32]

    L. H. Rowen, Polynomial Identities in Ring Theory, Pure Appl. Math.84, Academic Press, New York, 1980.

  • [33]

    Y. Wang, Power-centralizing automorphisms of Lie ideals in prime rings, Comm. Algebra 34 (2006), no. 2, 609–615.

    • Crossref
    • Export Citation
  • [34]

    T.-L. Wong, Derivations with power-central values on multilinear polynomials, Algebra Colloq. 3 (1996), no. 4, 369–378.

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