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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThen $R$ is [[Artinian|ZV04WWYE2Y]] if and only if $R$ is [[Noetherian|0MVH4Y1CRZ]] and every [[prime ideal|06HC6HK0Z8]] of $R$ is [[maximal|09J6KGT65G]].\n\n## Remarks\n\nThe condition that every prime ideal is maximal says that $R$ has [[Krull dimension|MVGBS3J1P7]] $0$ when $R$ is not the zero ring.\nIf $R$ is Artinian, each prime $P$ gives an Artinian domain $R/P$, which is a field, so $P$ is maximal.\nThen $R$ has finitely many maximal ideals $\\mathfrak{m}_1, \\dots, \\mathfrak{m}_n$, and $(\\mathfrak{m}_1 \\cdots \\mathfrak{m}_n)^k = 0$ for some $k$.\nThe chain $R \\supseteq \\mathfrak{m}_1 \\supseteq \\mathfrak{m}_1 \\mathfrak{m}_2 \\supseteq \\cdots \\supseteq 0$ has factors that are vector spaces over fields $R/\\mathfrak{m}_i$, and for vector spaces the two chain conditions agree, so $R$ has finite [[length|CZSCNZH8VY]] and is Noetherian.\nFor example, $\\Z/12\\Z$ is Artinian and Noetherian, while $\\Z$ is Noetherian but has the non-maximal prime $(0)$, so it is not Artinian.\nFor noncommutative rings, the Hopkins–Levitzki theorem says that every left Artinian ring is left Noetherian, while the converse needs extra hypotheses.\n\n## Explanations\n\n### L1 human\n\nFor commutative rings, the decreasing chain condition is stronger than the increasing one.\nThe theorem says exactly how much stronger: Artinian means Noetherian plus \"no prime ideal sits strictly inside another\".\n\n### L2 human\n\nThe theorem shows that Artinian rings are the Noetherian rings of dimension $0$, and it is often used together with the [[structure theorem for Artinian rings|HE7MP4HP6Q]].\nA typical use: if $R$ is a Noetherian local ring and $I$ is an ideal whose radical is the maximal ideal, then $R/I$ is Artinian, so it has finite length.\nThis fact is the start of dimension theory and of the [[Krull–Akizuki theorem|FSXA3CF365]].\n",
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  "msc": [
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  "name": "Akizuki–Hopkins–Levitzki theorem",
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