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  "body": "\n## Summary\n\nA filtration of a [[module|0NZ8JA25WK]] $M$ is a decreasing sequence of [[submodules|05WDSN597C]] $M = M_0 \\supseteq M_1 \\supseteq M_2 \\supseteq \\cdots$.\nFor an ideal $I$, an $I$-filtration is one with $I M_n \\subseteq M_{n+1}$, and it is stable when equality holds from some point on.\nThe basic example is the $I$-adic filtration $M_n = I^n M$; filtrations lead to the [[associated graded ring|8HFAEGFQT1]], the [[$I$-adic topology|2VF2TSJ3PY]] and the [[Artin–Rees lemma|30H6B0G2FT]].\n\n## Explanations\n\n### L1 human\n\nA filtration sorts the elements of a module by how small they are.\nThe deeper the submodule $M_n$, the smaller its elements are thought to be.\nFor the integers with the prime $p$, the submodule $p^n \\Z$ contains the integers divisible by $p^n$, and these count as small when $n$ is large.\nThe [[definition|GEJYZEY7Y0]] makes this precise.\n",
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