On a Quotient Ring That Satisfies Certain Identities via Generalized Reverse Derivations - Nawaf L. Alsowait; Mohammed Al-Shomrani; Radwan M. Al-omary; Zakia Z. Al-Amery (Mathematics, 2025)
In this article, for a prime ideal <semantics>ρ</semantics> of an arbitrary ring ℜ, we study the commutativity of the quotient ring <semantics>ℜ/ρ</semantics>, whenever ℜ admits a generalized reverse derivation <semantics>ϑ</semantics> associated with a reverse derivation ∂ that satisfies certain identities in <semantics>ρ</semantics>. Additionally, we show that, for some cases, the range of the generalized reverse derivation <semantics>ϑ</semantics> lies in the prime ideal <semantics>ρ</semantics>. 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.