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Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 95–119 | Cite as

Generalized derivations acting on multilinear polynomials in prime rings

  • Basudeb Dhara
Article

Abstract

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x1,..., x n ) a multilinear polynomial over C which is not central valued on R. If
$$F(f(r))G(f(r)) = H(f(r)^2 )$$
for all r = (r1,..., r n ) ∈ I n , then one of the following conditions holds:
  1. (1)

    there exist aC and bU such that F(x) = ax, G(x) = xb and H(x) = xab for all xR

     
  2. (2)

    there exist a, bU such that F(x) = xa, G(x) = bx and H(x) = abx for all xR, with abC

     
  3. (3)

    there exist bC and aU such that F(x) = ax, G(x) = bx and H(x) = abx for all xR

     
  4. (4)
    f(x1,..., x n )2 is central valued on R and one of the following conditions holds
    1. (a)

      there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all xR, with ab = p + p

       
    2. (b)

      there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all xR, with p + p’ = ab ∈ C.

       
     

MSC 2010

prime ring derivation generalized derivation extended centroid Utumi quotient ring 

MSC 2010

16W25 16N60 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Department of MathematicsBelda CollegeWest BengalIndia

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