relation between support and submodules

Question

Support of an $R$-module $M$ is defined as $\mathrm{Supp}_R(M)=\{\mathfrak{p}\in Spec(R); M_{\mathfrak{p}}\neq 0\}$. Let $M$ and $N$ two finitely generated modules over a Noetherian ring $R$ such that $\mathrm{Supp}_R(N)\subseteq \mathrm{Supp}_R(M)$. Is it true that $N$ is isomorphic to a submodule of $M$?

Answer 1

No. Take $R = \Bbb Z$ and consider the modules $M = (\Bbb Z/p)^2 \oplus \Bbb Z/q$ and $N=\Bbb Z/p \oplus (\Bbb Z/q)^2$. Then $\operatorname{Supp}(M)=\{(p),(q)\}=\operatorname{Supp}(N)$ but neither $M\subseteq N$ nor $N\subseteq M$.