baer and quasi-baer properties of skew pbw extensions

A ring r with an automorphism σ and a σ-derivation δ is called δ-quasi-baer (resp., σ-invariant quasi-baer) if the right annihilator of every δ-ideal (resp., σ-invariant ideal) of r is generated by an idempotent, as a right ideal. in this paper, we study baer and quasi-baer properties of skew pbw extensions. more exactly, let a=σ(r)⟨x1,…,xn⟩ be a skew pbw extension of derivation type of a ring r. (i) it is shown that r is δ-quasi-baer if and only if a is quasi-baer. (ii) r is δ-baer if and only if a is baer, when r has ifp. also, let a=σ(r)⟨x1,…,xn⟩ be a quasi-commutative skew pbw extension of a ring r. (iii) if r is a σ-quasi-baer ring, then a is a quasi-baer ring. (iv) if a is a quasi-baer ring, then r is a σ-invariant quasi-baer ring. (v) if r is a σ-baer ring, then a is a baer ring, when r has ifp. (vi) if a is a baer ring, then r is a σ-invariant baer ring. finally, we show that if a=σ(r)⟨x1,…,xn⟩ is a bijective skew pbw extension of a quasi-baer ring r, then a is a quasi-baer ring.