Let R be a prime ring of characteristic different from 2, Qr be the right Martindale quotient ring of R and C= Z(Qr) be the extended centroid of R. Suppose that f(x1, … , xn) is a noncentral multilinear polynomial over C and F, G are two nonzero generalized skew-derivations of R associated to the same automorphism of R. If F(u)u-G(u)F(u)=0for all u∈ f(R) , then one of the following holds: (1)there exist a, p∈ Qr such that F(x) = ax and G(x) = pxp- 1 for all x∈ R, with p- 1a∈ C;(2)there exist a, c, p∈ Qr such that F(x) = ax+ pxp- 1c and G(x) = pxp- 1 for all x∈ R, with f(R) 2⊆ C and p- 1(a- c) ∈ C;(3)there exist a, p∈ Qr such that F(x) = ax- pxp- 1a and G(x) = - pxp- 1 for all x∈ R, with f(R) 2⊆ C.
Generalized skew derivations and generalization of commuting maps on prime rings
De Filippis V.;
2022-01-01
Abstract
Let R be a prime ring of characteristic different from 2, Qr be the right Martindale quotient ring of R and C= Z(Qr) be the extended centroid of R. Suppose that f(x1, … , xn) is a noncentral multilinear polynomial over C and F, G are two nonzero generalized skew-derivations of R associated to the same automorphism of R. If F(u)u-G(u)F(u)=0for all u∈ f(R) , then one of the following holds: (1)there exist a, p∈ Qr such that F(x) = ax and G(x) = pxp- 1 for all x∈ R, with p- 1a∈ C;(2)there exist a, c, p∈ Qr such that F(x) = ax+ pxp- 1c and G(x) = pxp- 1 for all x∈ R, with f(R) 2⊆ C and p- 1(a- c) ∈ C;(3)there exist a, p∈ Qr such that F(x) = ax- pxp- 1a and G(x) = - pxp- 1 for all x∈ R, with f(R) 2⊆ C.Pubblicazioni consigliate
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