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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]].\n$R$ is *atomic* if every nonzero element $r$ of $R$ that is not a [[unit|019NFGFPR5]] has a factorization into irreducibles.\nThat is, $r = a_1 a_2 \\cdots a_n$ for some $n \\ge 1$ and some [[irreducible elements|052SDEPZH7]] $a_1, \\ldots, a_n$ of $R$.\n\n## Remarks\n\nExample: $\\Z[\\sqrt{-5}]$ is atomic, because it is Noetherian.\nIn a Noetherian domain, a nonzero nonunit without a factorization would give an infinite strictly ascending chain of principal ideals.\nNon-example: the ring $\\overline{\\Z}$ of all algebraic integers is not atomic.\nFor a nonzero nonunit $\\alpha$, a square root $\\beta$ of $\\alpha$ is an algebraic integer and a nonunit, and $\\alpha = \\beta^2$.\nSo no element of $\\overline{\\Z}$ is irreducible, and $2$ has no factorization.\nA domain that satisfies the ascending chain condition on principal ideals is atomic; the converse is false.\nThe factorization is not required to be unique; uniqueness up to order and units is what makes $R$ a [[unique factorisation domain|06YMXJMXYS]].\n",
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