Structure of (<i>σ, ρ</i>)-<i>n</i>-Derivations on Rings

The goal of this research is to describe the structure of Jordan (σ, ρ)-n-derivations on a prime ring. By (σ, ρ)- n-derivations, we mean n-additive maps ℑ : Rn →R satisfying the following property in each n-slot: ℑ(pq, ϖ1, ··· , ϖn−1) = ℑ(p, ϖ1, ··· , ϖn−1)σ(q) +ρ(p)ℑ(q, ϖ1, ··· , ϖn−1), for every p, q, ϖ1, ··· , ϖn−1∈ R. We find the conditions under which every Jordan (σ, ρ)-n-derivation becomes a (σ, ρ)-n-derivation. Moreover, the concept of ∗-n-centralizers on ∗-ring has given. The ∗-ring is also used for examining some outcomes, where left and right ∗-n-centralizers are significant