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  "body": "\n## Summary\n\nA complete local ring is a Noetherian [[local ring|MZ02BKPHJA]] $(R, \\mathfrak m)$ that equals its own [[completion|KBN13NPYJQ]] for the $\\mathfrak m$-adic topology.\nFormal power series rings over a field and the $p$-adic integers are the basic examples.\nComplete local rings have a clear structure, given by the [[Cohen structure theorem|N3XH42KVV3]], and every Noetherian local ring embeds into a complete one, its completion.\n\n## Explanations\n\n### L1 human\n\nIn a complete local ring, every list of better and better approximations has a limit.\nFor example, in the formal power series ring $k[\\![x]\\!]$ the polynomials $1$, $1 + x$, $1 + x + x^2$, $\\dots$ approach the series $1 + x + x^2 + \\cdots$, which is again an element of the ring.\nIn the ring of rational functions defined at $0$, many such lists have no limit, so that ring is not complete.\nThe [[definition|78GT1VFFPR]] makes this precise.\n",
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