Let R be a prime ring, Z (R) its center, U its right Utumi quotient ring, C its extended centroid, G a non-zero generalized derivation of R, f(x 1, ..., x n) a non-zero polynomial over C and I a non-zero right ideal of R. If f(x 1...,x n) is not central valued on R and [G(f(r 1..., r n)), f(r 1, ..., r n)] ∈C, for all r 1, ..., r n∈I, then either there exist a ∈ U, α ∈ C such that G(x) = ax for all x ∈ R, with (a;α)I = 0 or thereexists an idempotentelement e ∈ soc(RC) such that IC = eRC and one of the following holds: 1. f(x 1..., x n) is central valued in eRCe; 2. char(R) = 2 and eRCe satisfies the standard identity s 4 3. char(R) = 2 and f(x 1..., x n 2 is central valued in eRCe; 4. f(x 1..., x n) 2 is central valued in eRCe and there exist a, b ∈ U, α ∈ C such that G(x) = ax + xb, for all x ∈ R, with (a - b + α) I = 0.
Centralizing generalized derivations on polynomials in prime rings
DE FILIPPIS, Vincenzo
2012-01-01
Abstract
Let R be a prime ring, Z (R) its center, U its right Utumi quotient ring, C its extended centroid, G a non-zero generalized derivation of R, f(x 1, ..., x n) a non-zero polynomial over C and I a non-zero right ideal of R. If f(x 1...,x n) is not central valued on R and [G(f(r 1..., r n)), f(r 1, ..., r n)] ∈C, for all r 1, ..., r n∈I, then either there exist a ∈ U, α ∈ C such that G(x) = ax for all x ∈ R, with (a;α)I = 0 or thereexists an idempotentelement e ∈ soc(RC) such that IC = eRC and one of the following holds: 1. f(x 1..., x n) is central valued in eRCe; 2. char(R) = 2 and eRCe satisfies the standard identity s 4 3. char(R) = 2 and f(x 1..., x n 2 is central valued in eRCe; 4. f(x 1..., x n) 2 is central valued in eRCe and there exist a, b ∈ U, α ∈ C such that G(x) = ax + xb, for all x ∈ R, with (a - b + α) I = 0.Pubblicazioni consigliate
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