Prime Ideals and Homoderivations on Rings

In this paper, we aim to establish a new approach that involves characterizing the commutativity of a quotient ring L/P with homoderivations of L satisfying some algebraic identities involving the prime ideal P. In addition, some well-known results regarding the commutativity of prime rings have been developed for homoderivations of the rings. Some of the results obtained in this context are as follows: Let L be a ring, P a prime ideal of L and ξ a nonzero homoderivation of L. If any one of the following holds then ξ (L) ⊆P or L/P is commutative integral domain: i) ξ (μ₁, μ₂) ∈P, ii) ξ (μ₁ oμ₂) ∈P, iii) ξ (μ₁, μ₂) -μ₁, μ₂ ∈P, iv) ξ (μ₁ oμ₂) -μ₁ oμ₂∈P v) ξ (μ₁ μ₂) -ξ (μ₁) ξ (μ₂) ∈P, vi) ξ (μ₁ μ₂) -ξ (μ₂) ξ (μ₁) ∈P, vii) ξ (μ₁) ξ (μ₂) -μ₁, μ₂ ∈P, viii) ξ (μ₁) ξ (μ₂) -μ₁ oμ₂∈P, for all μ₁, μ₂∈ L.