I'm have been reading Einsenbud's commutative algebra.
And wanted to do exercise 2.9.
Exercise 2.9: One way of describing the ring $R\left[U^{-1}\right]$ is to say what its modules are: Show that an $R\left[U^{-1}\right]$ -module is the same thing as an $R$ module on which the elements of $U$ act as automorphisms. In particular, the map $M \rightarrow M\left[U^{-1}\right]$ is an isomorphism iff the elements of $U$ act as automorphisms on $M$.
And while I believe that I understand the idea of the problem, I'm not sure how to do it.
I understand that if $U$ acts as Automorphism on $M$ then $M$ will be a $R[U^{-1}]$ module, as $um\in M$ hence $umu^{-1}=m$. Any help will be appreciated.