Baer and Quasi-Baer Differential Polynomial Rings

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