Lemma 10.8.3. Let $(M_ i, \mu _{ij})$ be a system of $R$-modules over the preordered set $I$. Assume that $I$ is directed. The colimit of the system $(M_ i, \mu _{ij})$ is canonically isomorphic to the module $M$ defined as follows:

1. as a set let

$M = \left(\coprod \nolimits _{i \in I} M_ i\right)/\sim$

where for $m \in M_ i$ and $m' \in M_{i'}$ we have

$m \sim m' \Leftrightarrow \mu _{ij}(m) = \mu _{i'j}(m')\text{ for some }j \geq i, i'$
2. as an abelian group for $m \in M_ i$ and $m' \in M_{i'}$ we define the sum of the classes of $m$ and $m'$ in $M$ to be the class of $\mu _{ij}(m) + \mu _{i'j}(m')$ where $j \in I$ is any index with $i \leq j$ and $i' \leq j$, and

3. as an $R$-module define for $m \in M_ i$ and $x \in R$ the product of $x$ and the class of $m$ in $M$ to be the class of $xm$ in $M$.

The canonical maps $\phi _ i : M_ i \to M$ are induced by the canonical maps $M_ i \to \coprod _{i \in I} M_ i$.

Proof. Omitted. Compare with Categories, Section 4.19. $\square$

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