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.