Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x1, . . ., xn) a noncentral multilinear polynomial over C. If δ(G(f(r1, . . ., rn))f(r1, . . ., rn)) = 0 for all r1, . . ., rn ∈ R, then f(x1, ..., xn)2 is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.
Centralizers of Generalized Derivations on Multilinear Polynomials in Prime Rings
CARINI, Luisa;DE FILIPPIS, Vincenzo
2012-01-01
Abstract
Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x1, . . ., xn) a noncentral multilinear polynomial over C. If δ(G(f(r1, . . ., rn))f(r1, . . ., rn)) = 0 for all r1, . . ., rn ∈ R, then f(x1, ..., xn)2 is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.File in questo prodotto:
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