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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThe ring $R$ is Noetherian if the set of its ideals, ordered by inclusion, satisfies the [[ascending chain condition|0ZCAXY9563]].\nThat is, for every chain $I_1 \\subseteq I_2 \\subseteq \\dots$ of ideals of $R$, there is $N$ with $I_n = I_N$ for all $n \\ge N$.\n\n## Remarks\n\nThe ring $R$ is Noetherian if and only if every ideal of $R$ is [[finitely generated|030TZ0DQWT]].\nEvery field and every [[principal ideal domain|0RZYDVZ475]] is Noetherian.\nBy [[Hilbert's basis theorem|0YKA9J3XEE]], $R[x_1, \\dots, x_n]$ is Noetherian when $R$ is.\nThe polynomial ring in infinitely many variables $x_1, x_2, \\dots$ over a field is not Noetherian: $(x_1) \\subset (x_1, x_2) \\subset \\dots$ never stops.\nFor a noncommutative ring, one defines left and right Noetherian rings with left and right ideals.\n\n## Explanations\n\n### L1 human\n\nIn a Noetherian ring, each ideal is generated by a finite list of elements.\nThis makes many arguments work by induction, because a growing chain of ideals must stop.\nRings met in practice, like $\\mathbb{Z}[x, y]$, are Noetherian.\n",
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