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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].\nThe module $M$ is *finitely presented* if there are integers $m, n \\ge 0$ and an [[exact sequence|JB7ZBFBA9C]] of $R$-modules\n$$R^m \\xrightarrow{\\ \\varphi\\ } R^n \\xrightarrow{\\ \\pi\\ } M \\to 0.$$\nSuch a sequence is a *finite presentation* of $M$.\n\n## Remarks\n\nEquivalently, $M$ is [[finitely generated|05QY1VEBYD]] and the kernel of some surjection $R^n \\to M$ is finitely generated.\nBy Schanuel's lemma, the kernel of every surjection $R^n \\to M$ is then finitely generated.\nThe map $\\varphi$ is given by an $n \\times m$ matrix over $R$, so $M$ is the quotient of $R^n$ by the column span of a matrix.\nOver a [[Noetherian ring|0670T59HGE]], every finitely generated module is finitely presented, since submodules of $R^n$ are finitely generated.\n\nExample: over $R = \\Z$, the module $\\Z/6\\Z$ is finitely presented.\nThe sequence $\\Z \\xrightarrow{6} \\Z \\to \\Z/6\\Z \\to 0$, with the second map the reduction modulo $6$, is exact, because the image of multiplication by $6$ is $6\\Z$, which is the kernel of the reduction.\n\nNon-example: let $k$ be a field, $R = k[x_1, x_2, x_3, \\ldots]$ the polynomial ring in infinitely many variables, and $M = R/(x_1, x_2, \\ldots) \\cong k$.\nThen $M$ is finitely generated, by the class of $1$.\nThe kernel of $R \\to M$ is the ideal $(x_1, x_2, \\ldots)$, which is not finitely generated, so $M$ is not finitely presented.\n",
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