Let $alpha$ be an endomorphism and $delta$ an $alpha$-derivation of a ring $R$. In this paper we study the relationship between an $R$-module $M_R$ and the general polynomial module $M[x]$ over the skew polynomial ring $R[x;alpha,delta]$. We introduce the notions of skew-Armendariz modules and skew quasi-Armendariz modules which are generalizations of $alpha$-Armendariz modules and extend the classes of non-reduced skew-Armendariz modules. An equivalent characterization of an $alpha$-skew Armendariz module is given. Some properties of this generalization are established, and connections of properties of a skew-Armendariz module $M_R$ with those of $M[x]_{R[x;alpha,delta]}$ are investigated. As a consequence we extend and unify several known results related to Armendariz modules.
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