Direct limit of rings

Remarks

These operations do not depend on the choices, and they make $\varinjlim R_i$ a ring; the maps $\varphi_i \colon R_i \to \varinjlim R_i$, $a \mapsto [a]$, are ring homomorphisms with $\varphi_j \circ f_{ij} = \varphi_i$. It is the direct limit of the system in the category of rings: for every ring $S$ and homomorphisms $g_i \colon R_i \to S$ with $g_j \circ f_{ij} = g_i$, there is exactly one homomorphism $g \colon \varinjlim R_i \to S$ with $g \circ \varphi_i = g_i$. Example: let $I = \N$, $R_n = k[x_1, \dots, x_n]$ and let $f_{mn}$ be the inclusion for $m \le n$. The direct limit is the polynomial ring $k[x_1, x_2, \dots]$ in countably many variables, since each polynomial involves only finitely many variables. This ring is not Noetherian, although every $R_n$ is, as the chain of ideals $(x_1) \subset (x_1, x_2) \subset \cdots$ shows. Non-example: if $I$ is not directed, say two incomparable indices $1, 2$ with rings $\Z$ and $\Z/2$ and no common upper bound. Then the sum $[a] + [b]$ for $a \in R_1$ and $b \in R_2$ is not defined, and the construction does not give a ring. A direct limit of fields is a field, and a direct limit of local rings along local homomorphisms is a local ring.