Some results on σ-ideal of σ-prime ring

Abstract

Let R be a σ-prime ring with characteristic not 2, Z(R) be the center of R, I be a nonzero σ-ideal of R, α,β : R → R be two automorphisms, d be a nonzero (α, β)-derivation of R and h be a nonzero derivation of R. In the present paper, it is shown that (i) If d (I) ⊂ Cα,β and β commutes with σ then R is commutative. (ii) Let α and β commute with σ. If a ∈ I ∩ Sσ (R) and [d(I),a]α,β ⊂ Cα,β then a ∈ Z(R). (iii) Let α,β and h commute with σ. If dh(I)⊂Cα,β and h(I)⊂I then R is commutative.