Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials. (28th October 2014) - De Filippis, V.; Scudo, G.; Sorrenti, L.
Abstract : Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R, f ( x 1, …, x n ) a noncentral multilinear polynomial over C in n noncommuting variables, and a, b ∈ R such that a [ F ( f ( r 1, …, r n ) ), f ( r 1, …, r n ) ] b = 0 for any r 1, …, r n ∈ R . Then one of the following holds: (1) a = 0 ; (2) b = 0 ; (3) there exists λ ∈ C such that F ( x ) = λ x, for all x ∈ R ; (4) there exist q ∈ U and λ ∈ C such that F ( x ) = ( q + λ ) x + x q, for all x ∈ R, and f ( x 1, …, x n ) 2 is central valued on R ; (5) there exist q ∈ U and λ, μ ∈ C such that F ( x ) = ( q + λ ) x + x q, for all x ∈ R, and a q = μ a, q b = μ b .