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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThe ring $R$ is *coherent* if every [[finitely generated ideal|030TZ0DQWT]] of $R$ is [[finitely presented|3EH4C3M5GV]] as an $R$-[[module|09VDS2W1E8]].\n\n## Remarks\n\nEquivalently, for every $a_1, \\dots, a_n \\in R$, the module of relations $\\set{(r_1, \\dots, r_n) \\in R^n : r_1 a_1 + \\cdots + r_n a_n = 0}$ is finitely generated.\nExample: every [[Noetherian ring|0MVH4Y1CRZ]] is coherent, because the module of relations is a submodule of $R^n$, which is a Noetherian module.\nNon-example: let $R = k[x, y_1, y_2, \\dots]/(x y_1, x y_2, \\dots)$ over a field $k$.\nThe relations of the single element $x$ form the ideal $(y_1, y_2, \\dots)$ of $R$, which is not finitely generated.\nSo the ideal $(x)$ is finitely generated but not finitely presented.\nFor a noncommutative ring one defines left coherent rings: every finitely generated left ideal is finitely presented.\n\n## Explanations\n\n### L1 human\n\nA ring is coherent when every finite list of elements has a finite list of basic relations, from which all relations follow.\n",
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