On a Quotient Ring That Satisfies Certain Identities via Generalized Reverse Derivations

In this article, for a prime ideal ρ of an arbitrary ring ℜ, we study the commutativity of the quotient ring ℜ/ρ, whenever ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation ∂ that satisfies certain identities in ρ. Additionally, we show that, for some cases, the range of the generalized reverse derivation ϑ lies in the prime ideal ρ. Moreover, we explore several consequences and special cases. Throughout, we provide examples to demonstrate that various restrictions in the assumptions of our results are essential.