TITLE: Is direct limit of local rings a local ring?
QUESTION [9 upvotes]: Let $\{R_i\}_{i\in A}$ be a directed set of commutative local rings with directed index set $A$, and let $R$ be the direct limit of this set. I want to know if $R$ is a local ring (we know that $R$ is a commutative ring).

REPLY [3 votes]: A probably cleaner answer:
Let $(R_i,\mathfrak{m}_i)$ be a direct system of local rings. To each of these we get an exact sequence
$$0\rightarrow \mathfrak{m}_i\rightarrow R_i\rightarrow R_i/\mathfrak{m}_i\rightarrow 0$$
and by applying the exact $\varinjlim$ functor we obtain a sequence $$0\rightarrow \varinjlim \mathfrak{m}_i\rightarrow \varinjlim R_i\rightarrow \varinjlim R_i/\mathfrak{m}_i\rightarrow 0.$$
The nonzero object on the rightmost side is a field (direct limit of fields is a field - choose an element, it comes from a field, so has an inverse) hence the ideal $\varinjlim \mathfrak{m}_i$ is a maximal ideal. It's certainly the only maximal ideal, as every other ideal is contained in it, which completes the proof.