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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThe ring $R$ is *Artinian* if the set of its [[ideals|0MYSA6MYNA]], ordered by inclusion, satisfies the descending chain condition.\nThat is, for every chain $I_1 \\supseteq I_2 \\supseteq \\cdots$ of ideals of $R$, there is an index $k$ with $I_n = I_k$ for all $n \\ge k$.\n\n## Remarks\n\nExample: $\\Z/n\\Z$ for $n \\ge 1$ is Artinian, because it has only finitely many ideals, so a decreasing chain takes only finitely many values.\nExample: $k[x]/(x^2)$ over a field $k$ is Artinian, because every ideal is a $k$-subspace of a $2$-dimensional space.\nNon-example: $\\Z$ is not Artinian, because $(2) \\supsetneq (4) \\supsetneq (8) \\supsetneq \\cdots$ never stops.\nEquivalently, $R$ is Artinian when it is an [[Artinian module|YQNZTQGP1X]] over itself, or when every nonempty set of ideals has a minimal element.\nFor a noncommutative ring one defines left Artinian and right Artinian rings with left and right ideals.\n\n## Explanations\n\n### L1 human\n\nAn Artinian ring is a ring with no endless decreasing chain of ideals.\nFields and finite rings are the simplest examples.\n",
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