On X-generalized skew derivations and their applications

Abstract

Let [Formula: see text] be a prime ring of characteristic different from [Formula: see text], [Formula: see text] be its right Martindale quotient ring and [Formula: see text] be its extended centroid, [Formula: see text] be an automorphism of [Formula: see text], [Formula: see text] be a skew derivation of [Formula: see text] with associated automorphism [Formula: see text], [Formula: see text] and [Formula: see text] be two nonzero [Formula: see text]-generalized skew derivation of [Formula: see text] with associated term [Formula: see text] and [Formula: see text], respectively, [Formula: see text] be the set of the evaluations of [Formula: see text] on [Formula: see text], where [Formula: see text] is a non-central multilinear polynomial over [Formula: see text] in [Formula: see text] non-commuting variables. Let [Formula: see text] be such that [Formula: see text] for all [Formula: see text]. Then one of the following statements holds: (a) [Formula: see text] is central valued on [Formula: see text] and there exist [Formula: see text] such that [Formula: see text], [Formula: see text] for any [Formula: see text] with [Formula: see text]; (b) [Formula: see text] and [Formula: see text]. In the last part of the paper, we present some applications on the basis of the foregoing proposed result.