Semiprime Ideal of Rings with Symmetric Bi-Derivations

Let Ω be a ring with ℘ a semiprime ideal of Ω, I an ideal of Ω, Δ ∶ Ω × Ω → Ω a symmetric bi-derivation and δ be the trace of Δ. In the present paper, we shall prove that δ is a ℘-commuting map on I if any one of the following holds: i. δ(σ) ○ κ ∈ ℘, ii. δ([σ, κ]) ± [δ(σ), κ] ∈ ℘, iii. δ(σ○κ)±(δ(σ)○κ) ∈ ℘, iv. δ([σ, κ])±δ(σ)○κ ∈ ℘, v. δ(σ○κ)±[δ(σ), κ] ∈ ℘, vi. δ(σ)○κ±[δ(κ), σ] ∈ ℘, vii. δ([σ, κ]) ± δ(σ) ○ κ − [δ(κ), σ] ∈ ℘, viii. δ([σ, κ]) ± [δ(σ), κ] + [δ(κ), σ] ∈ ℘, ix. Δ(σ, κκ3) ±Δ(σ, κ)κ3 ∈ ℘, x. Δ(δ(σ), σ) ∈ ℘, xi. δ(δ(σ)) = g(σ), xii. δ(σ)κ ± σg(κ) ∈ ℘, xiii. [δ(σ), κ] ± [g(κ), σ] ∈ ℘, xiv. δ(σ) ○ κ ± (σ ○ g(κ)) ∈ ℘, xv. [δ(σ), κ] ± (σ ○ g(κ)) ∈ ℘, xvi. δ(σ) ○ κ ± [g(κ), σ] ∈ ℘ for all σ, κ ∈ I where G ∶ ℵ × ℵ → ℵ is a symmetric bi-derivation such that g is the trace of G.