Exploring Commutativity via Generalized (α, β)-Derivations Involving Prime Ideals

The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring ℑ/℘, where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized (α,β)-derivation associated with an (α,β)-derivation ∂ that satisfies certain criteria involving ℘, where α and β are automorphisms on ℑ. Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results.