Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x1, . . . , xn) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x1, . . . , xn) on R. If F and G are generalized derivations of R such that [F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds: (1) there exists α ∈ C such that F(x) = αx for all x ∈ R; (2) there exists β ∈ C such that G(x) = βx for all x ∈ R; (3) f(x1, . . . , xn)2 is central valued on R and either there exist a ∈ U and α ∈ C such that F(x) = ax + xa + αx for all x ∈ R or there exist c ∈ U and β ∈ C such that G(x) = cx + xc + βx for all x ∈ R; (4) R satisfies the standard identity s4(x1, . . . , x4) and either there exist a ∈ U and α ∈ C such that F(x) = ax + xa + αx for all x ∈ R or there exist c ∈ U and β ∈ C such that G(x) = cx + xc + βx for all x ∈ R.
Multilinear polynomials and cocentralizing conditions in prime rings
RANIA F
2011-01-01
Abstract
Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x1, . . . , xn) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x1, . . . , xn) on R. If F and G are generalized derivations of R such that [F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds: (1) there exists α ∈ C such that F(x) = αx for all x ∈ R; (2) there exists β ∈ C such that G(x) = βx for all x ∈ R; (3) f(x1, . . . , xn)2 is central valued on R and either there exist a ∈ U and α ∈ C such that F(x) = ax + xa + αx for all x ∈ R or there exist c ∈ U and β ∈ C such that G(x) = cx + xc + βx for all x ∈ R; (4) R satisfies the standard identity s4(x1, . . . , x4) and either there exist a ∈ U and α ∈ C such that F(x) = ax + xa + αx for all x ∈ R or there exist c ∈ U and β ∈ C such that G(x) = cx + xc + βx for all x ∈ R.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.