Journal of the
Korean Mathematical Society JKMS

For an endomorphism $\alpha$ of a ring $R$, we introduce the weak $\alpha$-skew Armendariz rings which are a generalization of the $\alpha$-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring $R$ is weak $\alpha$-skew Armendariz if and only if for any $n$, the $n\times n$ upper triangular matrix ring $T_n(R)$ is weak $\bar{\alpha}$-skew Armendariz, where $\bar{\alpha}:T_n(R)\rightarrow T_n(R)$ is an extension of $\alpha$. If $R$ is reversible and $\alpha$ satisfies the condition that $ab=0$ implies $a\alpha(b)$=0 for any $a,b\in R$, then the ring $R[x]/(x^n)$ is weak $\bar{\alpha}$-skew Armendariz, where $(x^n)$ is an ideal generated by $x^n$, $n$ is a positive integer and $\bar{\alpha}:R[x]/(x^n)\rightarrow R[x]/(x^n)$ is an extension of $\alpha$. If $\alpha$ also satisfies the condition that $\alpha^t=1$ for some positive integer $t$, the ring $R[x]$ (resp, $R[x;\alpha]$) is weak $\bar{\alpha}$-skew (resp, weak) Armendariz, where $\bar{\alpha}:R[x]\rightarrow R[x]$ is an extension of $\alpha$.

Keywords: reversible rings, $\alpha$-skew Armendariz rings, weak Armendariz rings, weak $\alpha$-skew Armendariz rings

MSC numbers: 16N60, 16P60