Let $A$ be a noetherian local ring with maximal ideal $m$. One says that an $A$-module $X$ is discrete if for every $x\in X$, there is a natural number $n$ such that $m^n.x=0$. 


My question is: Given an $A$-module $M$, a submodule $N\subseteq M$ and a submodule $P\subseteq N$ such that $N/P$ is discrete, is there a submodule $Q\subseteq M$ such that $M/Q$ is discrete and $Q\cap N\subseteq P$?


In fact, the title of my question is the same as the title of §7.2 of Kato's paper "A generalization of local class field theory by using K-groups. I". 


In that paper, the standard topology on an $A$-module $M$ is the group topology defined by taking as a base of open neighborhoods of $0\in M$ all $A$-submodules $N$ such that $M/N$ are discrete. 


Kato then wrote :


"If $N$ is an $A$-submodule of $M$, the standard topology on $N$ over $A$ coincides with the restriction of the standard topology of $M$ over $A$."


I think this statement is equivalent to a positive answer to my question above. It may be a quite elementary argument but I'm just stuck. So, could anyone give some help?



The standard topology of a module over a noetherian local ring
The assertion is indeed true in a more general context. My references are B. Stenström: *Rings of Quotients*, and Lam: *Lectures on Modules and Rings*.


Let a (right) linear topology on a ring $A$ be given by a family of (right) ideals $\mathfrak{F}$ (i.e. $\mathfrak{F}$ consists of all open right ideals of $A$). On any right $A$-module $M$ define the $\mathfrak{F}$-topology to be the one where a submodule $L \subseteq M$ is open iff for every $x \in M$ there is $I \in \mathfrak{F}$ such that $x I \subseteq L$.


In our setting, we have $\mathfrak{F} = \{ \text{ideals which contain some } \mathfrak{m}^n , n \in \mathbb{N} \}$.


Stenström Ch. VI Proposition 7.1 (due to Gabriel) gives equivalent criteria for a topology (given as $\mathfrak{F}$ as above) to satisfy what we want, namely:


$(\*)$ Given a submodule $N \subseteq M$, the $\mathfrak{F}$-topology on $N$ is the subspace topology of the $\mathfrak{F}$-topology on $M$.


In Stenström, an $\mathfrak{F}$ with this property is called *stable*. One of the equivalent criteria is 


$(\*\*)$ If a module $M$ is $\mathfrak{F}$-discrete, then so is its injective hull $E(M)$.


To see that $(\*\*) \Rightarrow (\*)$, take like you did $P \subseteq N$ such that $N/P$ is $\mathfrak{F}$-discrete; we have to show that there is a submodule $Q \subseteq M$ such that $M/Q$ is $\mathfrak{F}$-discrete and $Q \cap N \subseteq P$. Indeed, with Zorn's lemma take a submodule $Q \subseteq M$ maximal with respect to the property $Q \cap N = P$. Now by maximality of $Q$ one can see that $M/Q$ is an essential extension of $(N + Q)/Q$, which in turn is $\mathfrak{F}$-discrete because it is isomorphic to $N/P$. So $(\*\*)$ implies that $M/Q$ is $\mathfrak{F}$-discrete (because as essential extension, it "is" a submodule of the injective hull, and we obviously have that submodules of discrete modules are discrete).


Now Stenström Ch. VII Theorem 4.4 and Proposition 4.5 show that if our ground ring $A$ is commutative and Noetherian, indeed *every Gabriel topology on $A$ is stable*. (A Gabriel topology is a topology $\mathfrak{F}$ as above which additionally satisfies: Given any right ideal $\mathfrak{a}$, if there is $\mathfrak{b} \in \mathfrak{F}$ such that $(\mathfrak{a} : b) \in \mathfrak{F}$ for all $b \in \mathfrak{b}$, then $\mathfrak{a} \in \mathfrak{F}$. Using finite generation of the $\mathfrak{m}^n$, our $\mathfrak{F}$ is a Gabriel topology.)


To show that we indeed have $(\*\*)$, one can use e.g. that for every indecompasable injective module $E$ with associated prime $\mathfrak{p}$, and every $x \in E$, there is a natural number $n$ such that $x \mathfrak{p}^n = 0 $. (A fact due to Noether herself, see Lam, Theorem 3.78). The associated primes of a module and any of its essential extensions (in particular the injective hull) coincide, and the associated primes of an $\mathfrak{F}$-discrete module are easily seen to be in $\mathfrak{F}$. We can conclude by writing our injective hull in question as direct sum of indecomposables.


