On Ideals and Behavior of Quotient Rings via Generalized (α,β)-Derivations

The purpose of this paper is to examine the behavior of a factor ring R/P with a partial range I of a ring R, where P is a prime ideal of R and I is a non-zero ideal of R such that P⊊I. In order to accomplish this objective, we will utilize specific algebraic identities that involve P and are related to generalized (α,β)-derivations F and G, associated with (α,β)-derivations d and g, respectively, where α and β are automorphisms of R. Additionally, we will list some important related consequences. Finally, we will provide an illustrative example to emphasize the importance of the hypotheses imposed in our theorems.