Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, Z(R) the center of R, f(x_i) a non-central multilinear polynomial over K, d and g derivations of R, a and b fixed elements of R. Denote f(R) the set of all evaluations of the polynomial f(x_i) in R. If a[d(u),u]+[g(u),u]b=0, for any u in f(R), we prove that one of the following holds: either d=g=0; or d=0 and b=0; or g=0 and a=0; or a,b are central elements and ad+bg =0. We also examine some consequences of this result related to generalized derivations and we prove that: if d is a derivation of R and g a generalized derivation of R such that g([d(u),u])=0, for any u in f(R), then either g=0 or d=0.
On Some Generalized Identities with Derivations on Multilinear Polynomials
CARINI, Luisa;DE FILIPPIS, Vincenzo;
2010-01-01
Abstract
Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, Z(R) the center of R, f(x_i) a non-central multilinear polynomial over K, d and g derivations of R, a and b fixed elements of R. Denote f(R) the set of all evaluations of the polynomial f(x_i) in R. If a[d(u),u]+[g(u),u]b=0, for any u in f(R), we prove that one of the following holds: either d=g=0; or d=0 and b=0; or g=0 and a=0; or a,b are central elements and ad+bg =0. We also examine some consequences of this result related to generalized derivations and we prove that: if d is a derivation of R and g a generalized derivation of R such that g([d(u),u])=0, for any u in f(R), then either g=0 or d=0.Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
