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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?

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    $\begingroup$ Because every module is a directed limit of finitely-presented modules. $\endgroup$
    – Zhen Lin
    Mar 15, 2013 at 18:04
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    $\begingroup$ Zhen means directed colimit. $\endgroup$ Mar 15, 2013 at 18:06
  • $\begingroup$ @ZhenLin: Sounds like an answer to me $\endgroup$
    – Jim
    Mar 15, 2013 at 18:15
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    $\begingroup$ But why is Zhen Lin's statement correct (i.e. how does one create the direct limit)? $\endgroup$
    – neilme
    Mar 15, 2013 at 19:01
  • $\begingroup$ @neilme: That's an interesting point and i am not sure. $\endgroup$
    – Manos
    Mar 15, 2013 at 19:31

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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.

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