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  "body": "\n## Summary\n\nThe Krull dimension of a commutative ring is the largest length of a strictly increasing chain of [[prime ideals|00MHHZ9SNE]].\nIt is the algebraic notion of dimension: the coordinate ring of an affine variety has Krull dimension equal to the dimension of the variety.\nFields have dimension $0$, and principal ideal domains that are not fields, such as $\\Z$, have dimension $1$.\nThe polynomial ring in $n$ variables over a field has dimension $n$.\n\n## Explanations\n\n### L1 human\n\nPoints, curves and surfaces have dimensions $0$, $1$ and $2$.\nIn algebra, a point corresponds to a maximal ideal, and a bigger irreducible object corresponds to a smaller prime ideal.\nThe Krull dimension counts the longest staircase of prime ideals, each strictly inside the next.\nIn $k[x, y]$ the staircase $(0) \\subset (x) \\subset (x, y)$ goes from the whole plane, to a line, to a point, and has $2$ steps.\n",
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