On skew-commuting generalized skew derivations in prime and semiprime rings

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Let [Formula: see text] be a noncommutative prime ring of characteristic different from [Formula: see text], with right Martindale quotient ring [Formula: see text] and extended centroid [Formula: see text]. Let [Formula: see text] be fixed integers and [Formula: see text] a nonzero generalized skew derivation of [Formula: see text]. In this paper, we investigate the set [Formula: see text] and prove that its left annihilator in [Formula: see text] is identically zero. Using the above result, we also study the identity [Formula: see text] for semiprime rings.