I am taking a special case here. 

Let $R$ be an $E_\infty$-ring spectrum. In [HA][1], Lurie proves we have a forgetful functor (part of monadic adjunction)
$$ U_R:Mod_R(Sp) \rightarrow Sp$$
where $Sp$ is in the $\infty$-category of spectra. 

$U_R$ reflects equivalences. But **is $U_R$ faithful**?


--- 
One categoriacally, $U$ is faithful in many cases, i.e. if we replace $Sp$ with $Ab$. 
Perhaps the answer is false in $\infty$-categories. 
 I'd like to understand what goes wrong. Some comments on the following would be helpful: 

 - A counter example where $U_R$ is not faithful. (i.e. is it faithful when $R=H\Bbb Z$? )
- A brief/reference explaination for what accounts of this. 

  [1]: http://people.math.harvard.edu/~lurie/papers/HA.pdf