Maps preserving triple product on rings

Let 𝑅 and 𝑅 be two unital rings such that 𝑅 contains a nontrivial idempotent 𝑃1. If 𝑅 is a prime ring, we characterize the form of bijective map ℓ : 𝑅→ 𝑅 which satises ℓ(𝐴𝐵𝑃) = ℓ(𝐴)ℓ(𝐵)ℓ(𝑃), for every 𝐴,𝐵 ∈ 𝑅 and 𝑃 {𝑃1; 𝑃2}, where 𝑃2 := 𝐼- 𝑃1 and 𝐼 is the unit member of 𝑅. It is shown that ’ is an isomorphism multiplied by a central element. Finally, wecharacterize the form of ℓ : 𝑅→ 𝑅 which satises ℓ(𝑃)ℓ(𝐴)ℓ(𝑃) = 𝑃𝐴𝑃, for every 𝐴 ∈ 𝑅 and 𝑃 {𝑃1; 𝑃2}.