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  "body": "\n## Summary\n\nA $C^\\infty$-ring is a set on which every [[smooth function|0V33BNKZ9S]] $f \\colon \\R^n \\to \\R$ acts as an $n$-ary operation, compatibly with composition.\nThe basic example is the ring $C^\\infty(M)$ of smooth functions on a [[smooth manifold|F45VQ47GPV]], where the operations are composition with $f$.\nEvery $C^\\infty$-ring is in particular a commutative $\\R$-algebra, but it also knows how to apply $\\exp$, $\\sin$ and all other smooth functions.\n$C^\\infty$-rings play for smooth manifolds the role that commutative rings play in algebraic geometry, and they give models of synthetic differential geometry.\n\n## Explanations\n\n### L1 human\n\nIn a ring one can add and multiply, so one can plug elements into polynomials.\nIn a $C^\\infty$-ring one can plug elements into any smooth function, such as $e^x$ or $\\sin(x + y)$.\nFor functions on a manifold this is natural: if $a$ and $b$ are smooth functions, so is $\\sin(a + b)$.\nRings of germs and rings such as $\\R[\\varepsilon]/(\\varepsilon^2)$, with $\\varepsilon^2 = 0$, are also $C^\\infty$-rings, and they let one treat infinitesimal points algebraically.\n",
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  "name": "C-infinity ring",
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