On prime ideals in rings with involution involving pair of generalized derivations

The overall goal of this research is to explore the interactions that exist between the structure of a quotient ring [Formula: see text] and the behavior of generalized derivations of an arbitrary ring with involution [Formula: see text]. To be more exact, we use two generalized derivations of [Formula: see text] that satisfy algebraic identities involving the prime ideal [Formula: see text] to describe the commutativity of [Formula: see text]. Besides, we offer instances to demonstrate that the multiple constraints necessary in the assumptions of our theorems are not excessive.