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  "body": "\n## Summary\n\nA coherent ring is a ring in which every finitely generated ideal is [[finitely presented|3EH4C3M5GV]] as a module.\nEvery [[Noetherian ring|0670T59HGE]] is coherent, but coherence is weaker, and it is kept in many constructions that lose the Noetherian property.\nOver a coherent ring, the finitely presented modules form an abelian category, which makes it a good setting for homological algebra without the Noetherian hypothesis.\n\n## Explanations\n\n### L1 human\n\nIn a Noetherian ring every ideal has finitely many generators, and the relations among the generators are also finitely generated.\nA coherent ring keeps only the second half for finitely generated ideals: once an ideal has finitely many generators, it has finitely many relations among them too.\nFor example, the polynomial ring $k[x_1, x_2, x_3, \\dots]$ in infinitely many variables over a field $k$ is coherent but not Noetherian.\n",
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