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  "body": "\n## Statement\n\nLet $R$ be a [[semiprimary ring|XVP517QRH5]]: its [[Jacobson radical|XA0DCNH7WY]] $J = J(R)$ is a [[nilpotent ideal|C2H26BYWHE]] and $R/J$ is a [[semisimple ring|NY25HXA478]].\nLet $M$ be a right $R$-module.\nThen the following conditions are equivalent.\n\n1. $M$ is [[Noetherian|AK1ZRCXR32]].\n2. $M$ is [[Artinian|YQNZTQGP1X]].\n3. $M$ has a [[composition series|0XV2A49X6Q]].\n\n## Remarks\n\nOver any ring, condition 3 holds exactly when $M$ is both Noetherian and Artinian, so the point is that each chain condition alone gives the other.\nChoose $n$ with $J^n = 0$ and look at the filtration $M \\supseteq MJ \\supseteq MJ^2 \\supseteq \\cdots \\supseteq MJ^n = 0$.\nEach factor $MJ^{i}/MJ^{i+1}$ is killed by $J$, so it is a module over the semisimple ring $R/J$, hence a [[semisimple module|N3MRHDKDGC]].\nA semisimple module satisfies one chain condition only when it is a finite direct sum of simple modules, and then it has finite length.\nSo each factor has finite length, and so does $M$.\n\nMain corollary: a right [[Artinian ring|20QEAM8AX2]] is semiprimary, because [[its Jacobson radical is nilpotent|H7RAXJV9CP]] and $R/J$ is Artinian and semiprimitive, hence semisimple.\nSo every right Artinian ring is right [[Noetherian|0670T59HGE]], and the same holds on the left.\n\nThe hypothesis on $R$ cannot be dropped.\nThe ring $\\Z$ is not semiprimary, and the Prüfer group $\\Z[1/p]/\\Z$ is an Artinian $\\Z$-module that is not Noetherian.\nAlso $\\Z$ itself is a Noetherian $\\Z$-module that is not Artinian.\n\nFor commutative rings, the [[Akizuki–Hopkins–Levitzki theorem|C7X657TF19]] gives the sharper description of Artinian rings as the Noetherian rings in which every prime ideal is maximal.\n\n## Explanations\n\n### L1 human\n\nFor modules in general, \"no infinite descending chains\" and \"no infinite ascending chains\" are different conditions.\nOver a semiprimary ring the two conditions agree, and both mean that the module has finite length.\nThe best-known case is the ring itself: a ring with the descending chain condition on right ideals also has the ascending one.\n\n### L2 human\n\nThe proof peels the module into finitely many layers using the powers of the radical.\nEach layer is a module over a semisimple ring, where both chain conditions just say \"finitely many simple summands\".\nThis argument is the reason why right Artinian rings are right Noetherian, a fact used all through the structure theory of Artinian rings, together with the [[Artin–Wedderburn theorem|R1F271180N]].\nThe converse fails: $\\Z$ is Noetherian but not Artinian, so Artinian rings are a strictly smaller class.\n",
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