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I'm reading Milne's book of algebraic geometry and he gives the following criterion for direct limits:

An $R$-module $M$ together with $R$-linear maps $\alpha^i: M_i \to M$ is the direct limit of a system $(M_i,\alpha^i_j)$ if and only if

$i)$ $M = \cup_{i \in I} \alpha^i (M_i)$

$ii)$ $m_i \in M_i$ maps to zero in $M$ if and only if it maps to zero in $M_j$ for some $j \geqslant i$, and

$iii)$ $\alpha^i = \alpha^j \circ \alpha^i_j$ for all $j \geqslant i$.

I could prove some parts until now, but not all.

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On what parts are you stuck? – Fredrik Meyer Dec 9 '12 at 23:37
I have to prove the reciprocal and finish to prove $ii)$. – Diego Silvera Dec 9 '12 at 23:43
I'm sorry, which part are you stuck on? – Alex Youcis Dec 10 '12 at 0:00
Again, I have to prove the reciprocal, that's to say, if $M$ holds $i)$ and $ii)$ then $M$ is the direct limit and I have to prove that if $M$ is the direct limit of above then $M$ holds condition $ii)$, becuase I've proven that follows $i$. – Diego Silvera Dec 10 '12 at 3:01
Hi Diego! I don't know how Milne defines direct limits, but it seems to me that this criterion is the content of lemma 5.30 of Rotman's "An Introduction to Homological Algebra", 2nd. ed., p. 243. – lentic catachresis Dec 10 '12 at 11:03

You need to add the condition $\alpha^i=\alpha^j\circ \alpha^i_j$ to the criterion.

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It may not be necessary, but I added it anyway. – Diego Silvera Dec 10 '12 at 10:52

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