Exercise 2.14

A partially ordered set $I$ is said to be a directed set if for each pair $i,j$ in $I$ there exists $k\in I$ such that $i\le k$ and $j\le k$.

Let $A$ be a ring, let $I$ be a directed set and let ${(M_i)}_{i\in I}$ be a family of $A$-modules indexed by $I$. For each pair $i,j$ in $I$ such that $i\le j$, let ${\mu }_{\mathit{ij}}:M_i\to M_j$ be an $A$-homomorphism, and suppose that the following axioms are satisfied:

1.

${\mu }_{\mathit{ii}}$ is the identity mapping of $M_i$, for all $i\in I$;

2.

${\mu }_{\mathit{ik}}={\mu }_{\mathit{jk}}\circ {\mu }_{\mathit{ij}}$ whenever $i\le j\le k$.

Then the modules $M_i$ and homomorphisms ${\mu }_{\mathit{ij}}$ are said to form a direct system $M=(M_i,{\mu }_{\mathit{ij}})$ over the directed set $I$.

We shall construct an $A$-module $M$ called the direct limit of the direct system $\mathbf{\text{M}}$. Let $C$ be the direct sum of the $M_i$, and identify each module $M_i$ with its canonical image in $C$. Let $D$ be the submodule of $C$ generated by all elements of the form $x_i-{\mu }_{\mathit{ij}}(x_i)$ where $i\le j$ and $x_i\in M_i$. Let $M=C∕D$, let $\mu :C\to M$ be the projection and let ${\mu }_i$ be the restriction of $\mu$ to $M_i$.

The module $M$, or more correctly the pair consisting of $M$ and the family of homomorphisms ${\mu }_i:M_i\to M$, is called the direct limit of the direct system $M$, and is written $\underset{\to }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{M}_i$. From the construction it is clear that ${\mu }_i={\mu }_j\circ {\mu }_{\mathit{ij}}$ whenever $i\le j$.