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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nThe poset $P$ satisfies the *descending chain condition* when every [[sequence|0G9EKM0E15]] $x_1 \\ge x_2 \\ge x_3 \\ge \\cdots$ in $P$ is eventually constant.\nThat is, there is an index $n$ such that $x_m = x_n$ for all $m \\ge n$.\n\n## Remarks\n\nExample: the ideals of the ring $\\Z / 12\\Z$, ordered by inclusion, satisfy the descending chain condition.\nCheck: there are only six ideals, so a decreasing sequence of ideals takes finitely many values and must be eventually constant.\nRings whose ideals satisfy the condition are the [[Artinian rings|20QEAM8AX2]].\n\nNon-example: the ideals of $\\Z$ do not satisfy it, since $2\\Z \\supsetneq 4\\Z \\supsetneq 8\\Z \\supsetneq \\cdots$ never stops.\n\nThe condition is the [[ascending chain condition|0GZ6Y1H4RN]] for the opposite order.\nA poset satisfies it if and only if every nonempty subset has a minimal element, that is, the order is [[well-founded|55WD0H03CZ]].\nThe direction from the descending chain condition to minimal elements uses the axiom of dependent choice.\nThe abbreviation DCC is common, and the minimal-element form is called the minimal condition.\n",
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