Generalized derivations acting on multilinear polynomials in prime rings |
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Authors: | Basudeb Dhara |
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Institution: | 1.Department of Mathematics,Belda College,West Bengal,India |
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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)
there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R - (2)
there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C - (3)
there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R - (4)
f( x1,..., x n ) 2 is central valued on R and one of the following conditions holds - (a)
there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’ - (b)
there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
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