A product of two generalized derivations on polynomials in prime rings

Abstract

LetR be a prime ring of characteristic different from 2,U the Utumi quotient ring ofR, C the extended centroid ofR, F andG non-zero generalized derivations ofR andf(x ₁, ...,x _n ) a polynomial overC. Denote byf(R) the set {f(r₁, ..., r_n): r₁, ..., r_n ∃ R} of all the evaluations off(x ₁, ...,x _n ) inR. Suppose thatf(x ₁, ...,x _n ) is not central valued onR. IfR does not embed inM ₂(K), the algebra of 2 × 2 matrices over a fieldK, and the composition (FG) acts as a generalized derivation on the elements off(R), then (FG) is a generalized derivation of R and one of the following holds:

• 1. there existsα ∈ C such thatF(x)=αx, for allx ∈ R;

• 2. there existsα ∈ C such thatG(x)=αx, for allx ∈ R;

• 3. there exista; b ∈ U such thatF(x)=ax, G(x)=bx, for allx ∈ R;

• 4. there exista; b ∈ U such thatF(x)=xa, G(x)=xb, for allx ∈ R;

• 5. there exista; b ∈ U, α,β ∈ C such thatF(x)=ax+xb, G(x)=αx+β(αx − xb), for allx ∈ R.