family of free rank-one $R$-modules $$\pi: \xi = P \times_{G} R \to X$$ parametrized by X; conversely to such a family $\xi$ we can attach its $G$-torsor of trivializations $$P = \xi^{\times} = \{ z \in \xi | z \text{ is an $R$-module
    generator of }\xi_{\pi (z)} \}$$ An obvious $R$-module associated to this situation, at least if $X$ is discrete, is $$\label{eq:31}
   \bigoplus_{x\in X} \xi_{x} {\cong}\operatorname*{colim}_{x\in X} \xi_{x}$$ If $X$ is discrete then $P$ is the $G$-space $$P = \coprod_{x\in X} P_{x},$$ and we can also form the $R$-module $$\label{eq:32}
    {\mathbb{Z}}[P]\otimes_{{\mathbb{Z}}[G]} R.$$ Here ${\mathbb{Z}}[P]$ is the free abelian group on the points of $P,$ ${\mathbb{Z}}[G]$ is the group ring of $G,$ and the natural map $${\mathbb{Z}}[G] \to R$$ is the counit of the adjunction $$\label{eq:29}
\xymatrix{ {{\mathbb{Z}}: {(\mbox{groups})}} \ar@<.3ex>[r] & {{(\mbox{rings})}:
{GL_{1}}.}  \ar@<.3ex>[l] }$$ In fact the two $R$-modules and are isomorphic, as they represent the same functor: $$\begin{aligned}
 \label{eq:56}
   {(\mbox{$R$--modules})} \bigl( \bigoplus_{x\in X} \xi_{x},T\bigr) &
{\cong}\prod_{x\in X} {(\mbox{$R$--modules})} (\xi_{x},T) \\
   & {\cong}\notag
   \prod_{x\in X} {(\mbox{$G$--sets})} (\xi_{x}^{\times},T)\\
   & {\cong}\notag
   {(\mbox{$G$--sets})} (P, T) \\
   & {\cong}\notag
      {(\mbox{${\mathbb{Z}}[G]$--modules})} ({\mathbb{Z}}[P],T) \\
   & {\cong}{(\mbox{$R$--modules})}({\mathbb{Z}}[P]\otimes_{{\mathbb{Z}}[G]}R,T). \notag\end{aligned}$$

These constructions generalize to give equivalent notions of $A$-module Thom spectrum when $A$ is an ${A_{\infty}}$ ring spectrum. 