Journal of the
Korean Mathematical Society JKMS

Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring $R$ is called {\it quasi-Armendariz} if $a_iRb_j=0$ for each $i,j$ whenever polynomials $f(x)=\sum_{i=0}^ma_ix^i, g(x)=\sum_{j=0}^nb_jx^j\in R[x]$ satisfy $f(x)R[x]g(x)=0$. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if $R$ is a semiprime ring with an epimorphism $\sigma$, then $f(x)R[x;\sigma]g(x)=0$ implies $a_iR\sigma^{i+k}(b_j)=0$ for any integer $k\geq 0$ and $i, j$, where $f(x)= \sum_{i=0}^ma_ix^i, g(x)= \sum_{j=0}^nb_jx^j\in R[x;\sigma]$. Moreover, we extend this property to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define $\sigma$-skew quasi-Armendariz rings for an endomorphism $\sigma$ of a ring $R$. Then we study several extensions of $\sigma$-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and $\sigma$-skew Armendariz rings.

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

MSC numbers: 16N60, 16S36