Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Because every module is a directed limit of finitely-presented modules. – Zhen Lin Mar 15 '13 at 18:04
Zhen means directed colimit. – Martin Brandenburg Mar 15 '13 at 18:06
@ZhenLin: Sounds like an answer to me – Jim Mar 15 '13 at 18:15
Yup, please make it an answer and i will accept it. – Manos Mar 15 '13 at 18:46
But why is Zhen Lin's statement correct (i.e. how does one create the direct limit)? – neilme Mar 15 '13 at 19:01
up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.