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  "body": "\n## Summary\n\nThe direct limit of a directed system of [[algebras|QJGPT2W0FT]] is the algebra obtained by gluing the algebras of the system along its [[homomorphisms|KBXSZC6BV1]].\nIt is the [[direct limit|X9HEQ29Q49]] in the category of algebras of a fixed signature, and it is the union when the maps are inclusions.\nVarieties are closed under direct limits, and every algebra is the direct limit of its finitely generated subalgebras.\nThe dual construction is the [[inverse limit of algebras|DE2KEQ85ZT]].\n\n## Explanations\n\n### L1 human\n\nImagine a growing sequence of algebras, each mapped into the next one.\nThe direct limit is what you get \"at the end of the sequence\".\nAn element lives in some stage, and two elements are equal if they become equal at some later stage.\nFor example, the groups $\\Z/p\\Z \\to \\Z/p^2\\Z \\to \\Z/p^3\\Z \\to \\cdots$, with each map $x \\mapsto px$, have as direct limit the Prüfer $p$-group, the group of all complex roots of unity of $p$-power order.\nThe rationals $\\Q$ under addition are the direct limit of the groups $\\frac{1}{n!}\\Z$, which is simply their union.\n",
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