On Commutativity of Rings with (σ, τ)-Biderivations | ||
| Journal of Al-Nahrain University - Science | ||
| Article 1, Volume 19, Issue 2, June 2016, Pages 124-129 | ||
| Author | ||
| Auday Hekmat Mahmood | ||
| 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. | ||
| Keywords | ||
| Prime rings; Automorphisms; Biderivation | ||
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