Difference between revisions of "Ascending chain condition"

Revision as of 20:13, 10 April 2009

Let $S$ be a partially ordered set. We say that $S$ satisfies the ascending chain condition (ACC) if every ascending chain $x_0 \leqslant x_1 \leqslant x_2 \leqslant \dotsc$ eventually stabilizes; that is, there is some $N\ge 0$ such that $x_n = x_N$ for all $n\ge N$ .

Similarly, if every descending chain $x_0 \geqslant x_1 \geqslant x_2 \geqslant \dotsc$ stabilizes, we say that $S$ satisfies the descending chain condition (DCC). A set $S$ with an ordering $\leqslant$ satisifes ACC if and only if its opposite ordering satisfies DCC.

Every finite ordered set necessarily satisfies both ACC and DCC.

Let $A$ be a ring, and let $M$ be an $A$ -module. If the set of sub-modules of $M$ with the ordering of $M$ satifies ACC, we say that $M$ is Noetherian. If this set satisfies DCC, we say that $M$ is Artinian.

Theorem. A partially ordered set $S$ satisfies the ascending chain condition if and only if every subset of $S$ has a maximal element.

Proof. First, suppose that every subset of $S$ has a maximal element. Then every ascending chain in $S$ has a maximal element, so $S$ satisfies ACC.

Now, suppose that some subset of $S$ has no maximal element. Then we can recursively define elements $x_0, x_1, \dotsc$ such that $x_{n+1} > x_n$ , for all $n\ge 0$ . This sequence constitutes an ascending chain that does not stabilize, so $S$ does not satisfy ACC. $\blacksquare$

This article is a stub. Help us out by expanding it.

@@ Line 16: / Line 16: @@
 say that <math>M</math> is [[Noetherian]].  If this set satisfies DCC, we say
 that <math>M</math> is [[Artinian]].
+'''Theorem.''' A partially ordered set <math>S</math> satisfies the ascending
+chain condition if and only if every subset of <math>S</math> has a
+[[maximal element]].
+''Proof.''  First, suppose that every subset of <math>S</math> has a maximal
+element.  Then every ascending chain in <math>S</math> has a maximal element,
+so <math>S</math> satisfies ACC.
+Now, suppose that some subset of <math>S</math> has no maximal element.  Then
+we can recursively define elements <math>x_0, x_1, \dotsc</math> such that
+<math>x_{n+1} > x_n</math>, for all <math>n\ge 0</math>.  This sequence constitutes
+an ascending chain that does not stabilize, so <math>S</math> does not
+satisfy ACC.  <math>\blacksquare</math>
 {{stub}}

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Difference between revisions of "Ascending chain condition"

Revision as of 20:13, 10 April 2009

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