• Journal of the Korean Mathematical Society
• /
• v.47 no.3
• /
• pp.455-466
• /
• 2010

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 $\alpha$.

• Journal of the Korean Mathematical Society
• /
• v.42 no.2
• /
• pp.353-363
• /
• 2005

For a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$, we introduce ($\alpha$, $\delta$)-skew Armendariz rings which are a generalization of $\alpha$-rigid rings and Armendariz rings, and investigate their properties. A semi prime left Goldie ring is $\alpha$-weak Armendariz if and only if it is $\alpha$-rigid. Moreover, we study on the relationship between the Baerness and p.p. property of a ring R and these of the skew polynomial ring R[x; $\alpha$, $\delta$] in case R is ($\alpha$, $\delta$)-skew Armendariz. As a consequence we obtain a generalization of [11], [14] and [16].