Pure Submodules and Finitely Presented versus Finitely Generated Submodules

Question

Let $A$ be a ring $M$ an $A$-module and $N$ a submodule. Definition: $N$ is called a pure submodule of $M$ if the sequence $0 \rightarrow N \otimes E \rightarrow M \otimes E$ is exact for every $A$-module $E$.

Since exactness commutes with inductive limits and since every module is the inductive limit of finitely generated submodules, then it is enough for a submodule $N$ to be pure, that the above sequence is exact for every finitely generated module $E$.

Question: Matsumura in his Commutative Ring Theory p. 53 comments that it is enough to restrict attention to finitely presented modules $E$. Since the set of finitely presented modules is a proper subset of the set of finitely generated modules over $A$, why is Matsumura's statement valid?

Answer 1

Fix a presentation of $M$, say generators $\{ g_i : i \in I \}$ and relations $\{ r_j : j \in J \}$. Consider, for each finite subset $I' \subseteq I$ and each finite subset $J' \subseteq J$, the module $M_{I', J'}$ generated by $\{ g_i : i \in I' \}$ modulo the relations $\{ r_j : j \in J' \}$, where we agree to ignore a relation if it involves generators not in $I'$. These modules are finitely presented by construction and fit into an obvious directed system, whose colimit is (isomorphic to) the module $M$ in a canonical way. (Send each generator to the corresponding element of $M$!)

Do note that the transition maps in this directed system are not injective; but this doesn't change the fact that directed colimits are exact.