BAER AND QUASI-BAER PROPERTIES OF SKEW PBW EXTENSIONS

Abstract

A ring $R$ with an automorphism $sigma$ and a $sigma$-derivation $delta$ is called $delta$-quasi-Baer (resp., $sigma$-invariant quasi-Baer) if the right annihilator of every $delta$-ideal (resp., $sigma$-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=sigma(R)leftlangle x_{1},ldots,x_{n}rightrangle $ be a skew PBW extension of derivation type of a ring $R$. (i) It is shown that $ R$ is $Delta$-quasi-Baer if and only if $ A$ is quasi-Baer.(ii) $ R$ is $Delta$-Baer if and only if $ A$ is Baer, when $R$ has IFP. Also, let $A=sigma (R)leftlangle x_1, ldots , x_nrightrangle$ be a quasi-commutative skew PBW extension of a ring $R$. (iii) If $R$ is a $Sigma$-quasi-Baer ring, then $A $ is a quasi-Baer ring. (iv) If $A $ is a quasi-Baer ring, then $R$ is a $Sigma$-invariant quasi-Baer ring. (v) If $R$ is a $Sigma$-Baer ring, then $A $ is a Baer ring, when $R$ has IFP. (vi) If $A $ is a Baer ring, then $R$ is a $Sigma$-invariant Baer ring. Finally, we show that if $A = sigma (R)leftlangle x_1, ldots , x_nrightrangle $ is a bijective skew PBW extension of a quasi-Baer ring $R$, then $A$ is a quasi-Baer ring.