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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.\nFor $m \\ge n$ let $\\pi_{mn} \\colon R/I^m \\to R/I^n$ be the natural map of [[quotient rings|0076HWHGHQ]].\nThe *$I$-adic completion* of $R$ is the inverse limit\n$$\\hat R = \\varprojlim_n R/I^n = \\set{(x_n)_{n \\ge 1} \\in \\prod_{n \\ge 1} R/I^n : \\pi_{mn}(x_m) = x_n \\text{ for all } m \\ge n},$$\nwith componentwise addition and multiplication.\n\n## Remarks\n\nThe map $R \\to \\hat R$, $r \\mapsto (r + I^n)_n$, is a ring homomorphism with kernel $\\bigcap_n I^n$.\nEquivalently, $\\hat R$ is the ring of Cauchy sequences for the [[$I$-adic topology|2VF2TSJ3PY]] modulo the sequences that tend to $0$.\nFor an $R$-module $M$ one defines $\\hat M = \\varprojlim_n M/I^n M$ in the same way.\nExample: for $R = k[x]$, with $k$ a field, and $I = (x)$, a compatible sequence is the list of truncations of a formal power series, and $\\hat R \\cong k[\\![x]\\!]$.\nNon-example: for $R = \\Z$ and $I = 2\\Z$, the sequence starting with $x_1 = 0 + 2\\Z$ and $x_2 = 1 + 4\\Z$ does not give an element of $\\hat R$, since $\\pi_{21}(x_2) = 1 + 2\\Z \\neq x_1$.\n",
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