a fibration of $*$-modules $$f: B\to B{{GL_{1}}R},$$ ${{GL_{1}}R}$ acts on the pull-back $P$ in the diagram $$\xymatrix{
P \ar[r] \ar[d] & E{{GL_{1}}R} \ar[d] \\
B \ar[r]^f & B{{GL_{1}}R},
}$$ and the $S$-module ${{\Sigma^{\infty}}_{+}}P$ is a right ${{\Sigma^{\infty}}_{+}}{{GL_{1}}R}$-module. We can then imitate to form an $R$-module Thom spectrum.

\[def-6\] Given a map of spaces $f \colon B \to B{{GL_{1}}R}$, form $P$ as the pullback in the diagram above associated to a fibrant replacement in $*$-modules of the free map $* {\times_{\mathcal{L}}}{\mathbb{L}}B \to B{{GL_{1}}R}$. The Thom spectrum of $f$ is defined to be the derived smash product $$\label{eq:62}
   Mf = {{\Sigma^{\infty}}_{+}}P {\wedge}^{L}_{{{\Sigma^{\infty}}_{+}}{{GL_{1}}R}} R.$$

With this definition, we have $$\label{eq:35}
   {(\mbox{right $R$--modules})} (M, R) {\simeq}{(\mbox{right
${{GL_{1}}R}$--spaces})} (P,{\Omega^{\infty}}R),$$ where here (and in the remainder of this subsection) we are referring to derived mapping spaces.

The space of *orientations* of $M$ is the subspace of $R$-module maps $M \to R$ which correspond to $${(\mbox{right ${{GL_{1}}R}$--modules})} (P,{{GL_{1}}R}) \subset {(\mbox{right
${{GL_{1}}R}$--modules})} (P,{\Omega^{\infty}}R).$$ under the weak equivalence . That is, we have a homotopy pull-back diagram $$\xymatrix{ {{(\mbox{orientations})} (M,R)} \ar[r]^-{{\simeq}} \ar[d] &
{{(\mbox{right ${{GL_{1}}R}$--spaces})} (P,{{GL_{1}}R})} \ar[d]
\\
{{(\mbox{right $R$--modules})} (M, R)} \ar[r]^-{{\simeq}} & {{(\mbox{right
${{GL_{1}}R}$--modules})} (P,{\Omega^{\infty}}R).}  }$$

To make contact with classical notions of orientation, for $x\in B$ 