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.