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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 *Artinian* if the set of its [[submodules|0RGZ10DFPG]], ordered by inclusion, satisfies the descending chain condition.\nThat is, for every chain $N_1 \\supseteq N_2 \\supseteq \\cdots$ of submodules of $M$, there is an index $k$ with $N_n = N_k$ for all $n \\ge k$.\n\n## Remarks\n\nEquivalently, every nonempty set of submodules of $M$ has a minimal element for inclusion.\nExample: a finite-dimensional vector space $V$ over a field is Artinian, because a strictly decreasing chain of subspaces has at most $\\dim V + 1$ terms.\nNon-example: $\\Z$ as a $\\Z$-module is not Artinian, because $2\\Z \\supsetneq 4\\Z \\supsetneq 8\\Z \\supsetneq \\cdots$ never stops.\nExample: for a prime $p$, the Prüfer group $\\Z[1/p]/\\Z$ is an Artinian $\\Z$-module, since its proper subgroups form the chain $0 \\subset \\frac{1}{p}\\Z/\\Z \\subset \\frac{1}{p^2}\\Z/\\Z \\subset \\cdots$.\nThat chain also shows that it is not a [[Noetherian module|AK1ZRCXR32]].\nSubmodules, quotients and extensions of Artinian modules are Artinian.\n\n## Explanations\n\n### L1 human\n\nAn Artinian module is a module with no endless decreasing chain of submodules.\nIt is the mirror image of a Noetherian module.\n",
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