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

Abstract

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) ⊂C_(α,β)(ii) [ImF , I]_(α,β) =0 (iii) F(xω, y) = F(ωx, y)for all x, y, ω∈ I.(2) Suppose〖 F〗_1: R⟶R is a nonzero(σ, τ)-derivation and F_2:R×R⟶R is a (α, β)-Biderivation with ImF_2=R, If F_1 F_2(I, I)=0 then F_2=0.