On Commutativity of Rings with (𝛔, 𝛕)-Biderivations

Let R be a prime ring with characteristic different from 2, ℐ be a nonzero ideal of R. in this paper, for α,β,σ,τ as automorphisms of R, we present some results concerning the relationship between the commutativity of a ring and the existence of specific types of a (σ,τ)-Biderivation, we prove: (1) Suppose F:R×R⟶R is a nonzero(σ,τ)-Biderivation then R is a commutative ring if F satisfies one of the following conditions:(i) F(ℐ, ℐ) ⊂Cα,β(ii) [ImF , ℐ]α,β =0 (iii) F(xω, y) = F(ωx, y)for all x, y, ω∈ ℐ.(2) Suppose F1: R⟶R is a nonzero(σ, τ)-derivation and F2:R×R⟶R is a (α, β)-Biderivation with ImF2=R, If F1F2(ℐ, ℐ)=0 then F2=0.