In this paper, the authors explored functional identities involving endomorphisms ( \(\Gamma \) ) and antiautomorphisms ( \(\Xi \) ) on prime rings, focusing on their impact on commutativity. The results show that a prime ring \({\Re }\) , where the characteristic of \({\Re }\) is not equal to two, with an antiautomorphism \(\Xi \) that is non-linear over the center \(Z_{\Re }\) of the ring \({\Re }\) , and an endomorphism \(\Gamma \) satisfies commutativity under conditions such as \(\Gamma (\Xi (u)u) \mp [\Xi (u), u] \in Z_{\Re }\) , \(\Gamma (\Xi (u)u) \mp \Xi (u) \circ u \in Z_{\Re }\) , or \(\Gamma (\Xi (u)u) \mp u\Xi (u) \in Z_{\Re }\) . Additionally, when \(\Gamma \) is a non-identity endomorphism, commutativity follows if \(\Gamma (\Xi (u)u) - \Xi (u)u \in Z_{\Re }\) . Furthermore, \({\Re }\) is either commutative or can be embedded in ( \(2 \times 2\) ) matrices over a field if \(\Gamma (\Xi (u)u) + \Xi (u)u \in Z_{\Re }\) . Examples are provided to highlight the necessity of these conditions and to demonstrate the practical implications of the findings.
