Communications of the
Korean Mathematical Society CKMS

The concept of the quasi-Armendariz property of rings properly contains Armendariz rings and semiprime rings. In this paper, we extend the quasi-Armendariz property for a polynomial ring to the skew polynomial ring, hence we call such ring a $\sigma$-quasi-Armendariz ring for a ring endomorphism $\sigma$, and investigate its structures, several extensions and related properties. In particular, we study the semiprimeness and the quasi-Armendariz property between a ring $R$ and the skew polynomial ring $R[x;\sigma]$ of $R$, and so these provide us with an opportunity to study quasi-Armendariz rings and semiprime rings in a general setting, and several known results follow as consequences of our results.

Keywords: quasi-Armendariz property, skew polynomial ring, semiprime ring, rigid ring

MSC numbers: 16S36, 16N60, 16W20