---
tag: Q5T05ET56S
kind: concept
name: Direct limit of rings
msc: [16S10, 18A30, 13B02]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_limit
---

## Summary

The direct limit of rings glues a directed family of [[rings|0B1261B5ED]] $R_i$, connected by ring homomorphisms $R_i \to R_j$ for $i \le j$, into one ring.
This ring receives a map from each $R_i$.
It is the [[direct limit|X9HEQ29Q49]] taken in the category of rings, and it is computed on the underlying sets.
Every element of the limit comes from some $R_i$, and two elements are equal if they become equal in some later $R_k$.
Polynomial rings in infinitely many variables and many infinite-dimensional algebras arise this way.

## Explanations

### L1 human

Imagine a chain of rings, each sitting inside the next one, such as $k[x_1] \subseteq k[x_1, x_2] \subseteq k[x_1, x_2, x_3] \subseteq \cdots$.
The direct limit is the union of the chain, here the polynomial ring in infinitely many variables.
When the maps are not inclusions, elements of different rings are identified as soon as they become equal further along.
Each computation in the limit takes place in one of the rings of the family, since only finitely many elements are involved.