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  "body": "\n## Summary\n\nThe coordinate ring of an algebraic set $X \\subseteq k^n$ is the ring of polynomial functions on $X$.\nIt is the quotient of the polynomial ring by the ideal of $X$.\nIt carries the geometry of $X$: points, closed subsets and morphisms of affine varieties are read off from it.\nAlso $X$ is irreducible exactly when this ring is an integral domain.\n\n## Explanations\n\n### L1 human\n\nOn a set of points cut out by equations, two polynomials give the same function when their difference vanishes there.\nThe coordinate ring is the ring of these functions.\nOn the circle $x^2 + y^2 = 1$, the polynomials $x^2$ and $1 - y^2$ give the same function, so they are equal in the coordinate ring.\nOn the $x$-axis, every polynomial in $x$ and $y$ agrees with a polynomial in $x$ alone, so the coordinate ring is just $k[x]$.\n",
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