On Skew Quasi-Baer Rings

Abstract

A ring R with an automorphism α and an α-derivation δ is called (α,δ)-quasi-Baer (resp., quasi-Baer) if the right annihilator of every (α,δ)-ideal (resp. ideal) of R is generated by an idempotent, as a right ideal. We show the left-right symmetry of (α, δ)-quasi Baer condition and prove that a ring R is (α, δ)-quasi Baer if and only if R[x; α, δ] is α-quasi Baer if and only if R[x; α, δ] is -quasi Baer for every extended derivation of δ. When R is a ring with IFP, then R is (α, δ)-Baer if and only if R[x; α, δ] is α-Baer if and only if R[x; α, δ] is -Baer for every extended α-derivation on R[x; α, δ] of δ. A rich source of examples for (α, δ)-quasi Baer rings is provided.