Reference: Atiyah and Macdonald, Introduction to Commutative algebra, page 46
Let $A$ be a commutative ring with 1 and $M$ a $A$-module. Then we define $Supp(M)$ to be the set of prime ideals $\mathfrak p$ of $A$ such that $M_\mathfrak p \neq 0$.
It is known that if $M$ is finitely generated then $Supp(M)=V(Ann(M))$ where $Ann(M)=\{f\in A |~ f M=0 \}$ and $V(E)=\{\mathfrak p \in Spec(A)|~ \mathfrak p \supset E\}$ for a subset $E$ of $A$.
(1) What is the relationship between $Supp(M)$ and $ V(Ann(M))$ if we drop the finitely generated condition, and do we have good examples to illustrate this?
(2) Another question is about the "naturality" of the support. Given a ring homomorphism $f:A\to B$ and a finitely generated $A$-module $M$, then we have $$ Supp(B\otimes_A M) = f^{*~-1}(Supp(M)) $$ where $f^*: Spec B \to Spec A$ is induced by $f$.
I can only prove $ Supp(B\otimes_A M) \supset f^{*~-1}(Supp(M)) $ by using (1) and observing that $f(Ann(M))\subset Ann(B\otimes_A M)$.