Browse > Article
http://dx.doi.org/10.14317/jami.2021.073

A NOTE ON GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS  

RAZA, MOHD ARIF (Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University)
REHMAN, NADEEM UR (Department of Mathematics, Aligarh Muslim University)
GOTMARE, A.R. (GDM Arts, KRN Commerce and MD Science Colledge)
Publication Information
Journal of applied mathematics & informatics / v.39, no.1_2, 2021 , pp. 73-81 More about this Journal
Abstract
Let R be a prime ring, Qr be the right Martindale quotient ring and C be the extended centroid of R. If �� be a nonzero generalized skew derivation of R and f(x1, x2, ⋯, xn) be a multilinear polynomial over C such that (��(f(x1, x2, ⋯, xn)) - f(x1, x2, ⋯, xn)) ∈ C for all x1, x2, ⋯, xn ∈ R, then either f(x1, x2, ⋯, xn) is central valued on R or R satisfies the standard identity s4(x1, x2, x3, x4).
Keywords
Prime ring; Generalized skew derivation; Automorphism;
Citations & Related Records
연도 인용수 순위
  • Reference
1 T.L. Wong, Derivation with power central values on multilinear polynomials, Algebra Colloq. 3 (1996), 369-378.
2 M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math. 42 (2002), 3-8.   DOI
3 N. Argac, V. De Filippis and H.G. Inceboz, Generalized derivations with power central values on multilinear polynomials on right ideals, Rend. Sem. Mat. Univ. Padova 120 (2008), 59-71.   DOI
4 K.I. Beidar, W.S. Martindale III and A.V. Mikhalev, Rings with Generalized Identities, Pure and Applied Mathematics, Marcel Dekker 196, New York, 1996.
5 H.E. Bell and M.N. Daif, Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci. 15 (1992), 205-206.   DOI
6 J. Bergen, Automorphisms with unipotent values, Rend. Circ. Math. Palermo Series II Tomo 31 (1982), 226-232.   DOI
7 C.M. Chang, Power central values of derivations on multilinear polynomials, Taiwanese J. Math 7 (2003), 329-338.   DOI
8 J.C. Chang, On the identity h(x) = af(x) + g(x)b, Taiwanese J. Math 7 (2003), 103-113.   DOI
9 M.C. Chou and C.K. Liu, An Engel condition with skew derivation, Monatash Math 158 (2009), 259-270.   DOI
10 C.L. Chuang, Differential identities with automorphism and anti-automorphism-I, J. Algebra 149 (1992), 371-404.   DOI
11 C.L. Chuang, Differential identities with automorphism and anti-automorphism-II, J. Algebra 160 (1993), 130-171.   DOI
12 C.L. Chuang and T.K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005), 59-77.   DOI
13 V. De Filippis and O.M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40 (2012), 1918-1932.   DOI
14 T.K. Lee, Derivations with invertiable values on a multilinear polynomial, Proc. Amer. Math. Soc. 119 (1993), 1077-1083.   DOI
15 M. Hogan, A note on semiprime rings with deribvation, Int. J. Math. Math. Sci. 20 (1997), 413-415.   DOI
16 N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub., Rhode Island, 1964.
17 V.K. Kharchenko, Generalized identities with automorphisms, Algebra Logic 14 (1975), 132-148.   DOI
18 U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103.   DOI
19 W.S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584.   DOI
20 M.A. Quadri, M.S., Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. pure appl. Math. 34 (2003), 1393-1396.
21 C.L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), 723-728.   DOI
22 N. Rehman and M.A. Raza, On m-commuting mappings with skew derivations in prime rings, St. Petersburg Math. J. 27 (2016), 641-650.   DOI
23 N. Rehman and M.A. Raza, On ideals with skew derivations of prime ring, Miskolc Math. Notes 15 (2014), 717-724.   DOI
24 L. Rowen, Polynomial identities in ring thory, Pure and Applied Math. 84, Academic Press, Inc., New York London, 1980.