I am trying to solve the problem number 20 of chapter 4 from the book of Atiyah and Macdonald's Introduction to Commutative Algebra. I solved everythng but this: to prove the analogous of $a\subseteq r(a)$ (where $r(a)$ is the radical of an ideal $a$) for $r_M(N)$ where $M$ is a fixed A-module and $N$ a submodule of $M$.
$r_M(N)$ is defined as $r_M(N)=\{x\in A:x^qM\subseteq N \text{ for some } q<0\}$
This problem is just a matter of definition. Since $N$ and $r_M(N)$ are not the same kind of object, it does not make sense to say that "$N \subseteq r_M(N)$". But what canonical ideal of $A$ can we get from the submodule $N$? Precisely $(N:M) = \{x \in A : xM \subseteq N\}$. You can now directly see that $(N:M) \subseteq r_M(N)$ by looking at their definitions, which should remind you of the proof that $\mathfrak{a} \subseteq r(\mathfrak{a})$ in the case of ideals.