Direct limit of rings

Summary

The direct limit of rings glues a directed family of rings $R_i$, 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 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.