Factor Rings with Algebraic Identities via Generalized Derivations

The current article focuses on studying the behavior of a ring ℜ/Π when ℜ admits generalized derivations Ψ and Ω with associated derivations ϕ and δ, respectively. These derivations satisfy specific differential identities involving Π, where Π is a prime ideal of an arbitrary ring ℜ, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of Π in the hypotheses of our various theorems, we provide a list of examples.