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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be an $R$-[[module|0NZ8JA25WK]].\nThe *$I$-adic topology* on $M$ is the following [[topology|0CQHFPGJ45]].\nA subset $U \\subseteq M$ is open if and only if for every $x \\in U$ there is $n \\ge 0$ with $x + I^n M \\subseteq U$.\nThe $I$-adic topology on $R$ is the case $M = R$.\n\n## Remarks\n\nThe cosets $x + I^n M$, for $x \\in M$ and $n \\ge 0$, form a base of this topology; each of them is open and also closed.\nAddition on $M$ and multiplication $R \\times M \\to M$ are continuous, so $R$ becomes a topological ring and $M$ a topological module.\nThe topology is Hausdorff if and only if $\\bigcap_n I^n M = 0$; then $d(x, y) = 2^{-\\nu(x - y)}$, with $\\nu(z)$ the largest $n$ such that $z \\in I^n M$ and $\\nu(0) = \\infty$, is a metric for it.\nExample: on $\\Z$ with $I = (2)$, the sequence $2^n$ tends to $0$, since $2^m \\in 2^n \\Z$ for all $m \\ge n$.\nNon-example: the set $\\set{0}$ is not open in $\\Z$ for this topology, since no $2^n \\Z$ is contained in $\\set{0}$.\nSo the $2$-adic topology on $\\Z$ is not discrete, unlike the topology that $\\Z$ gets from $\\R$.\n",
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