Jordan derivations and generalized derivations in semiprime rings: Structural theorems and applications

Let [Formula: see text] be an associative semiprime ring with identity element [Formula: see text], and let [Formula: see text] be an additive mapping from [Formula: see text] to itself. The main goal of this paper is to establish specific identities involving [Formula: see text] in the context of an [Formula: see text] that is [Formula: see text]-torsion free for some integer [Formula: see text]. Consequently, this classification identifies [Formula: see text] as a derivation or a generalized derivation of [Formula: see text].