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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[module|09VDS2W1E8]] over $R$.\nThe module $M$ is *Noetherian* if the set of its [[submodules|0RGZ10DFPG]], ordered by inclusion, satisfies the [[ascending chain condition|0ZCAXY9563]].\nThat is, for every chain $N_1 \\subseteq N_2 \\subseteq \\cdots$ of submodules of $M$, there is an index $k$ with $N_n = N_k$ for all $n \\ge k$.\n\n## Remarks\n\nThe following conditions are equivalent: $M$ is Noetherian; every submodule of $M$ is [[finitely generated|0XHM30WBVB]]; every nonempty set of submodules of $M$ has a maximal element for inclusion.\nExample: $\\Z$ as a $\\Z$-module is Noetherian, because each submodule is $n\\Z$ for some $n$, generated by one element.\nNon-example: $\\Q$ as a $\\Z$-module is not Noetherian, because $\\Z \\subsetneq \\frac{1}{2}\\Z \\subsetneq \\frac{1}{4}\\Z \\subsetneq \\cdots$ is a chain of submodules that never stops.\nSubmodules and quotients of Noetherian modules are Noetherian, and an extension of two Noetherian modules is Noetherian.\nOver a [[Noetherian ring|0MVH4Y1CRZ]], every finitely generated module is Noetherian.\n\n## Explanations\n\n### L1 human\n\nA Noetherian module is a module with no endless increasing chain of submodules.\nEquivalently, each of its submodules can be described by finitely many generators.\n",
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