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| − | Let <math>S</math> be a [[partially ordered set]]. We say that <math>S</math> satisfies the '''ascending chain condition''' ('''ACC''') if every ascending chain
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| − | <cmath> x_0 \leqslant x_1 \leqslant x_2 \leqslant \dotsc </cmath>
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| − | eventually stabilizes; that is, there is some <math>N\ge 0</math> such that
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| − | <math>x_n = x_N</math> for all <math>n\ge N</math>.
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| − | Similarly, if every descending chain
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| − | <cmath> x_0 \geqslant x_1 \geqslant x_2 \geqslant \dotsc </cmath>
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| − | stabilizes, we say that <math>S</math> satisfies the '''descending chain condition''' ('''DCC'''). A set <math>S</math> with an ordering <math>\leqslant</math> satisfies ACC if and only if
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| − | its opposite ordering satisfies DCC.
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| − | Every [[finite]] ordered set necessarily satisfies both ACC and
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| − | DCC.
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| − | Let <math>A</math> be a [[ring]], and let <math>M</math> be an <math>A</math>-module. If the set
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| − | of sub-modules of <math>M</math> with the ordering of <math>M</math> satifies ACC, we
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| − | say that <math>M</math> is [[Noetherian]]. If this set satisfies DCC, we say
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| − | that <math>M</math> is [[Artinian]].
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| − | '''Theorem.''' A partially ordered set <math>S</math> satisfies the ascending
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| − | chain condition if and only if every subset of <math>S</math> has a
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| − | [[maximal element]].
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| − | ''Proof.'' First, suppose that every subset of <math>S</math> has a maximal
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| − | element. Then every ascending chain in <math>S</math> has a maximal element,
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| − | so <math>S</math> satisfies ACC.
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| − | Now, suppose that some subset of <math>S</math> has no maximal element. Then
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| − | we can recursively define elements <math>x_0, x_1, \dotsc</math> such that
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| − | <math>x_{n+1} > x_n</math>, for all <math>n\ge 0</math>. This sequence constitutes
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| − | an ascending chain that does not stabilize, so <math>S</math> does not
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| − | satisfy ACC. <math>\blacksquare</math>
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| − | {{stub}}
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| − | == See also ==
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| − | * [[Noetherian]]
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| − | * [[Artinian]]
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| − | * [[Partially ordered set]]
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| − | * [[Zorn's Lemma]]<!-- haha-->
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| − | [[Category:Set theory]]
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| − | [[Category:Ring theory]]
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