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  "body": "\n## Definition\n\nLet $(I, \\le)$ be a directed set: a nonempty preordered set in which any two elements have a common upper bound.\nA *directed system of rings* over $I$ consists of [[rings|0B1261B5ED]] $R_i$ for $i \\in I$ and [[ring homomorphisms|0VTH8KPCG2]] $f_{ij} \\colon R_i \\to R_j$ for $i \\le j$, such that:\n\n1. $f_{ii}$ is the identity of $R_i$ for every $i$.\n2. $f_{jk} \\circ f_{ij} = f_{ik}$ whenever $i \\le j \\le k$.\n\nOn the disjoint union $\\bigsqcup_i R_i$, let $a \\in R_i$ and $b \\in R_j$ be [[equivalent|0G8VEZ3WTK]] if $f_{ik}(a) = f_{jk}(b)$ for some $k$ with $i \\le k$ and $j \\le k$.\nThe *direct limit* $\\varinjlim R_i$ is the set of equivalence classes $[a]$.\nFor $a \\in R_i$ and $b \\in R_j$, choose $k \\ge i, j$ and set $[a] + [b] = [f_{ik}(a) + f_{jk}(b)]$ and $[a][b] = [f_{ik}(a) f_{jk}(b)]$.\n\n## Remarks\n\nThese 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 $\\varphi_j \\circ f_{ij} = \\varphi_i$.\nIt is the [[direct limit|X9HEQ29Q49]] 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 $g \\circ \\varphi_i = g_i$.\nExample: let $I = \\N$, $R_n = k[x_1, \\dots, x_n]$ and let $f_{mn}$ be the inclusion for $m \\le n$.\nThe direct limit is the polynomial ring $k[x_1, x_2, \\dots]$ in countably many variables, since each polynomial involves only finitely many variables.\nThis ring is not [[Noetherian|0670T59HGE]], although every $R_n$ is, as the chain of ideals $(x_1) \\subset (x_1, x_2) \\subset \\cdots$ shows.\nNon-example: if $I$ is not directed, say two incomparable indices $1, 2$ with rings $\\Z$ and $\\Z/2$ and no common upper bound.\nThen 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.\nA direct limit of fields is a field, and a direct limit of local rings along local homomorphisms is a local ring.\n",
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