Let M be a left R-module and R be a ring with unity, and $S={0,2,3,4,{ldots}}$ be a submonoid. Then $M[x^{-s}]={a_0+a_2x^{-2}+a_3x^{-3}+{cdots}+a_nx^{-n}{mid}a_i{in}M}$ is an $R[x^s]$-module. In this paper we show some properties of $M[x^{-s}]$ as an $R[x^s]$-module. Let $f:M{ ightarrow}N$ be an R-linear map and $overline{M}[x^{-s}]={a_2x^{-2}+a_3x^{-3}+{cdots}+a_nx^{-n}{mid}a_i{in}M}$ and define $N+overline{M}[x^{-s}]={b_0+a_2x^{-2}+a_3x^{-3}+{cdots}+a_nx^{-n}{mid}b_0{in}N,;a_i{in}M}$. Then $N+overline{M}[x^{-s}]$ is an $R[x^s]$-module. We show that given a short exact sequence $0{ ightarrow}L{ ightarrow}M{ ightarrow}N{ ightarrow}0$ of R-modules, $0{ ightarrow}L{ ightarrow}M[x^{-s}]{ ightarrow}N+overline{M}[x^{-s}]{ ightarrow}0$ is a short exact sequence of $R[x^s]$-module. Then we show $E_1+overline{E_0}[x^{-s}]$ is not an injective left $R[x^s]$-module, in general.