let $M_{x}$ be the Thom spectrum associated to the map $$\{x \} \to B {\xrightarrow{f}} B{{GL_{1}}R}.$$ The map $x \to B$ on passage to Thom spectra gives rise to a map of $R$-modules $M_{x} \to M$. Then we have the following (Theorem \[t-th-orientation-ainfty\] and Proposition \[t-pr-characterize-orientations\]).

\[t-th-or-unstable-and-stable\]

1.  A map of right $R$-modules $u: M\to R$ is an orientation if and only if for each $x\in B$, the map of $R$-modules $$M_{x} \to M \to R$$ is a weak equivalence.

2.  If $f: B\to B{{GL_{1}}R}$ is a fibration, then the space of orientations $M\to R$ is weakly equivalent to the derived space of lifts $$\xymatrix{ {P} \ar[r] \ar[d] & {E{{GL_{1}}R}} \ar[d]
    \\
    {B} \ar[r]^-{f} \ar@{-->}[ur] & {B{{GL_{1}}R}.}  }$$

As for the Thom isomorphism, just as in the classical situation we have an $R$-module Thom diagonal $$M \to {{\Sigma^{\infty}}_{+}}B {\wedge}M.$$ and given a map of right $R$-modules $u: M\to R$, we obtain the composite map of right $R$-modules $${\rho}(u): M \to {{\Sigma^{\infty}}_{+}}B {\wedge}M {\xrightarrow{1 {\wedge}f}} {{\Sigma^{\infty}}_{+}}B {\wedge}R,$$ about which we have the following (See [@MR609673] and [@LMS:esht §IX]).

If $u: M \to R$ is an orientation, then ${\rho}(u)$ is a weak equivalence.

As in the ${E_{\infty}}$ case, we emphasize that our 