Document Type : Original Manuscript

Authors

1 Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran.

2 Department of Mathematics, University of Yazd, P.O. Box 89195-741, Yazd, Iran.

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 $A=\sigma(R)\left\langle x_{1},\ldots,x_{n}\right\rangle$ 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 $A=\sigma (R)\left\langle x_1, \ldots , x_n\right\rangle$ 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 $A = \sigma (R)\left\langle x_1, \ldots , x_n\right\rangle$ is a bijective skew PBW extension of a quasi-Baer ring $R$, then $A$ is a quasi-Baer ring.

Keywords