This is a question about support of modules under extension of scalars.


Let $f \colon A \to B$ be a homomorphism of commutative rings (with unity), and let $M$ be a finitely generated $A$-module. 
Recall that the *support* of $M$ is the set of prime ideals $\mathfrak{p}$ of $A$ such that the localization $M\_{\mathfrak{p}}$ 
is nonzero.
Then 
$\mathrm{Supp} \_B(B \otimes\_A M) = f^{\*-1}(\mathrm{Supp}\_A(M))$, 
the set of prime ideals of $B$ whose contractions are in the support of $M$. 


The $\subseteq$ containment is true for any $M$.
What's an obvious example of a non-finitely generated module where the other containment doesn't hold?



Extension of scalars and support of a non-finitely generated module
Take $A = \mathbb Z$, $M = \mathbb Q$, $B = \mathbb Z/p\mathbb Z$, were $p$ is a prime.


