Is the direct limit of submodules also a submodule?

Question

$\{M_i\}_{i\in I}$ is a directed system with direct limit $M$. For each $i\in I$, $N_i\subseteq M_i$ is a submodule and $\{N_i\}_{i\in I}$ with the restriction maps is also a directed system with direct limit $N$. So there is a natural map from $N$ to $M$.

Is this map always injective? If the answer is no, what is the condition for the map to be injective?

This problem is not obvious to me. Consider the example: choose $x$ and $y$ from $M_i$, such that $x+y\in N_i$, $\mu_{i,j}(x)\in N_j$ and $\mu_{i,k}(y)\in N_k$ for some $j, k\ge i$ (assume they exist). Then obviously $$ x-\mu_{i,j}(x)+y-\mu_{i,k}(y)=0\in M. $$ Now how to prove that $x-\mu_{i,j}(x)+y-\mu_{i,k}(y)=0$ in $N$? Or such $x$ and $y$ do not exist at all?

I suggest you try to prove it and let us know where you get stuck (if at all). — Claudius, Mar 22 '18 at 09:22
You have to use that the set $I$ is directed. This is a very strong property. — Claudius, Mar 22 '18 at 09:39
You should consider writing up your solution as an answer (this is not frowned upon; in fact this is highly encouraged on this site). This will help later visitors who are wondering about the same thing. — Claudius, Mar 23 '18 at 12:12

score 2 · Answer 1 · answered Jul 23 '23 at 00:28

As a visitor who was wondering about the same question, here is my answer.

Let $\textbf{M} = ((M_i),(\mu_{ij}))$ be a direct system with direct limit $(M, (\mu_i))$. Suppose that for each $i$, there are sub-modules $N_i \le M_i$ such that $\mu_{ij}(N_i) \subseteq N_j$. Then, it is straightforward to verify that $ \textbf{N} =((N_i), (\mu_{ij} \vert_{N_i}))$ is also a direct system; let the direct limit be $(N, (\nu_i))$.

Rather than working with the troublesome construction of the direct limit, consider the family of inclusion homomorphisms $\iota_i : N_i \to M_i$. It is clear that $\iota_j \circ \mu_{ij} \vert_{N_i} = \mu_{ij} \circ\iota_i$, so this gives a homomorphism of direct systems $\textbf{M} \to \textbf{N}$, which in turn descends to a homomorphism $\iota : M \to N$ satisfying $\iota \circ \nu_i = \mu_i \circ \iota_i$.

We claim that $\iota$ is injective. Let $n \in N$. Then, $\exists i$ s.t. $\nu_i(n_i) = n$.

Suppose that $\iota(n)=0$.

$$\implies \iota(\nu_i(n_i)) = 0$$ $$\implies (\mu_i \circ \iota_i)(n_i) = 0$$ $$\implies \mu_i(n_i) = 0$$ $$\implies \exists j \; \text{ s.t. } \mu_{ij} (n_i) = 0$$ $$\implies \exists j \; \text{ s.t. } \mu_{ij} \vert_{N_i} (n_i) = 0$$

$$\implies \nu_i (n_i) = (\nu_j \circ \mu_{ij} \vert_{N_i} )(n_i) = 0$$

$$\implies n = 0$$

Hence, $\iota$ is injective, and $N$ can be identified as a sub-module of $M$.

Is the direct limit of submodules also a submodule?

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