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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]], let $M$ be an $R$-[[module|0NZ8JA25WK]], and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.\nA *filtration* of $M$ is a sequence of [[submodules|05WDSN597C]] $(M_n)_{n \\ge 0}$ with $M_0 = M$ and $M_n \\supseteq M_{n+1}$ for all $n$.\nIt is an *$I$-filtration* if moreover $I M_n \\subseteq M_{n+1}$ for all $n \\ge 0$.\nAn $I$-filtration is *stable* if there is $n_0$ such that $I M_n = M_{n+1}$ for all $n \\ge n_0$.\n\n## Remarks\n\nExample: the $I$-adic filtration $M_n = I^n M$ is a stable $I$-filtration, since $I \\cdot I^n M = I^{n+1} M$ for every $n$.\nNon-example: for $R = M = \\Z$ and $I = 2\\Z$, the filtration $M_n = 4^n \\Z$ is not a $2\\Z$-filtration, since $I M_1 = 8\\Z$ is not contained in $M_2 = 16\\Z$.\nSome authors say $I$-good for stable.\nTwo stable $I$-filtrations of $M$ have bounded difference: there is $k$ with $M_{n+k} \\subseteq M'_n$ and $M'_{n+k} \\subseteq M_n$ for all $n$, so they define the same topology.\n",
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