Asian Research Journal of Mathematics

This paper examines the interaction between nonzero (\(\vartheta\), \(\vartheta\))-derivations and nonzero \(\vartheta\)-centralizers in prime rings, where \(\vartheta\) is an automorphism. The study considers a sequence of algebraic identities involving derivations, centralizers, commutators, and the centre of the ring, and establishes sufficient conditions under which either the derivation vanishes, the ring is commutative, or the derivation is commuting. After recalling the definitions of derivations, (\(\theta\), \(\varphi\))-derivations, centralizers, and \(\theta\)-centralizers, a preliminary lemma is used to support the subsequent arguments. The main results show that several commutator relations between a (\(\vartheta\), \(\vartheta\))-derivation and a \(\vartheta\)-centralizer force commutativity of the prime ring or triviality of the derivation. Additional conditions involving images of commutators, products of the derivation and centralizer, and central elements are shown to imply that the derivation is commuting. Further identities involving the \(\vartheta\)-centralizer alone also yield commutativity. The proofs rely on primeness, the automorphism property of \(\vartheta\), standard commutator identities, substitutions into the assumed relations, and repeated use of the preliminary lemma. Collectively, the results provide a unified set of sufficient conditions connecting (\(\vartheta\), \(\vartheta\))-derivations and \(\vartheta\)-centralizers with commutativity properties in prime rings, while remaining within the algebraic framework specified in the manuscript. The argument proceeds theorem by theorem, preserving the stated assumptions on the mappings and using the same algebraic setting throughou.