A note on generalized skew-derivations on multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic different from 2, Q_r its right
Martindale quotient ring and C its extended centroid, f(X₁; : : : ;X_n) a multilinear
polynomial over C that is noncentral-valued on R and F a generalized skew derivation
of R. If for some 0 ≠ a ∈R-C,
a[F(f(r)); f(r)] - [F(f(r)); f(r)]α ∈ Z(R)
for all r = (r₁; : : : ; r_n) ∈ Rⁿ then one of the following conditions holds:
(1) there exists λ, ∈ C such that F(x) = λx for all x ∈ R;
(2) there exist b ∈ Q_r and λ ∈ C such that F(x) = bx + xb + λx for all x ∈ R and
f(X₁; : : : ;X_n)² is central valued on R.

As an application of this result, we investigate the commutator [S; T] ∈ Z(R), where
S = {[F(u); u]lu ∈ f(R)} and T = f[G(v); v]jv ∈ f(R)}, F and G two generalized
skew derivations of R.

Key words: Prime ring, derivation, generalized derivation, generalized skew derivation,
extended centroid.