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.