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([μ_1,μ_2 ])∈P, ii) ξ(μ_1 oμ_2)∈P, iii) ξ([μ_1,μ_2 ])-[μ_1,μ_2 ]∈P, iv) ξ(μ_1 oμ_2 )-μ_1 oμ_2∈P v) ξ(μ_1 μ_2)-ξ(μ_1)ξ(μ_2)∈P, vi) ξ(μ_1 μ_2)-ξ(μ_2) ξ(μ_1)∈P, vii) ξ(μ_1) ξ(μ_2)-[μ_1,μ_2 ]∈P, viii)ξ(μ_1) ξ(μ_2)-μ_1 oμ_2∈P, for all μ_1,μ_2∈ L.