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  "body": "\n## Summary\n\nThe completion of a [[commutative ring|0310REJJY8]] $R$ with respect to an [[ideal|096Z9MP2ME]] $I$ is the ring $\\hat R = \\varprojlim R/I^n$.\nIts elements are compatible sequences of residues modulo the powers of $I$.\nIt is the completion of $R$ for the [[$I$-adic topology|2VF2TSJ3PY]], in the same way as the real numbers complete the rationals.\nThe $p$-adic integers and the formal power series rings arise in this way.\nCompletion is a basic tool to study a local ring near its closed point.\n\n## Explanations\n\n### L1 human\n\nKnowing an integer modulo $2$, modulo $4$, modulo $8$, and so on, determines it.\nA compatible list of such residues need not come from an integer, and the completion adds all such lists as new elements.\nFor $\\Z$ and the prime $p$ one gets the [[$p$-adic integers|V6TJAFZ4MD]].\nFor $k[x]$ and the ideal $(x)$ one gets the [[formal power series|033WGN00BF]] $k[\\![x]\\!]$, where an element is known by its truncations modulo every power of $x$.\nThe [[definition|NS84D2920R]] makes this precise.\n",
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