Definition 10.8.1. Let $(I, \leq )$ be a preordered set. A system $(M_ i, \mu _{ij})$ of $R$-modules over $I$ consists of a family of $R$-modules $\{ M_ i\} _{i\in I}$ indexed by $I$ and a family of $R$-module maps $\{ \mu _{ij} : M_ i \to M_ j\} _{i \leq j}$ such that for all $i \leq j \leq k$

$\mu _{ii} = \text{id}_{M_ i}\quad \mu _{ik} = \mu _{jk}\circ \mu _{ij}$

We say $(M_ i, \mu _{ij})$ is a directed system if $I$ is a directed set.

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