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  "body": "\n## Summary\n\nDimension theory is the part of commutative algebra that studies the [[Krull dimension|MVGBS3J1P7]] of rings and modules.\nIt also studies related notions such as [[height|87WH65FJB8]], [[system of parameters|95QTEPMDA4]] and [[multiplicity|JX20MNDDXT]].\nIts central results are [[Krull's principal ideal theorem|SJZW2HN1SC]] and the [[dimension theorem for local rings|KW3HY77V6D]].\nThe latter computes the Krull dimension of a Noetherian local ring from its [[Hilbert–Samuel function|BNQVS83GZV]].\nIt gives an algebraic meaning to the dimension of an algebraic variety.\n\n## Explanations\n\n### L1 human\n\nA line has dimension $1$ and a plane has dimension $2$.\nDimension theory explains what dimension means for an arbitrary commutative ring, where there is no picture to look at.\nThe main idea is to count the longest chain of prime ideals, and then to show that this count agrees with other natural counts.\nFor example, the polynomial ring $k[x, y]$ over a field has the chain $0 \\subsetneq (x) \\subsetneq (x, y)$, and its dimension is $2$, as for the plane.\n",
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