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Keywords:
Armendariz ring; $(\alpha ,\delta )$-skew Armendariz ring; weak Armendariz ring; weak $(\alpha ,\delta )$-skew Armendariz ring
Summary:
In this note, for a ring endomorphism $\alpha $ and an $\alpha $-derivation $\delta $ of a ring $R$, the notion of weakened $(\alpha ,\delta )$-skew Armendariz rings is introduced as a generalization of $\alpha $-rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(\alpha ,\delta )$-skew Armendariz ring if and only if $T_n(R)$ is weakened $(\bar {\alpha },\bar {\delta })$-skew Armendariz if and only if $R[x]/(x^n)$ is weakened $(\bar {\alpha },\bar {\delta })$-skew Armendariz ring for any positive integer $n$.
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