Definition 111.2.1. A *directed set* is a nonempty set $I$ endowed with a preorder $\leq $ such that given any pair $i, j \in I$ there exists a $k \in I$ such that $i \leq k$ and $j \leq k$. A *system of rings* over $I$ is given by a ring $A_ i$ for each $i \in I$ and a map of rings $\varphi _{ij} : A_ i \to A_ j$ whenever $i \leq j$ such that the composition $A_ i \to A_ j \to A_ k$ is equal to $A_ i \to A_ k$ whenever $i \leq j \leq k$.

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