Successive localizations of a module

Question

I am trying to prove the following:

Suppose that $p\subseteq q$ are prime ideals in $R$ and $M$ is an $R$-module. Then the the localization of the $R$-module $M_{q}$ at $p$ is $M_{p}$, i.e., $(M_{q})_{p}=M_{p}$.

I have tried to use the fact that $M_{p}\simeq R_{p} \otimes M$ as R-modules but I couldn't get the statement. Any help would be great.

Answer 1

Hint: Let $A\to B$ a (commutative) $A$-algebra, $B\to C$ a $B$-algebra, $M$ an $A$-module. There is a canonical isomorphism: $$(N\otimes_A B)\otimes_B C\simeq M\otimes_A C.$$