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  "body": "\n## Summary\n\nThe length of a module is the largest number of strict steps in a chain of its submodules.\nIt measures the size of a module, as dimension measures the size of a vector space.\nA module has finite length exactly when it is both a [[Noetherian module|AK1ZRCXR32]] and an [[Artinian module|YQNZTQGP1X]].\nLength is additive along short exact sequences, which makes it a basic tool in dimension theory and multiplicity theory.\n\n## Explanations\n\n### L1 human\n\nClimb from the zero submodule to the whole module, one strict step at a time, and count the steps.\nThe length is the largest number of steps that you can take.\nFor a vector space this is its dimension.\nFor $\\Z/12\\Z$ as a $\\Z$-module the length is $3$, as in $0 \\subset 6\\Z/12\\Z \\subset 3\\Z/12\\Z \\subset \\Z/12\\Z$, matching the three prime factors of $12 = 2 \\cdot 2 \\cdot 3$.\n",
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