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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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    Hi Bruno! That works, thanks Bruno.2012-12-10

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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.2012-12-10