Difference between revisions of "Ascending chain condition"
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Revision as of 20:59, 10 April 2009
Let 
be a partially ordered set. We say that satisfies the ascending chain condition (ACC) if every ascending chain
![\[x_0 \leqslant x_1 \leqslant x_2 \leqslant \dotsc\]](http://latex.artofproblemsolving.com/d/9/2/d925945400c03b89ef547f81c1f58566f125fffd.png)
eventually stabilizes; that is, there is some 
such that
for all 
.
Similarly, if every descending chain
![\[x_0 \geqslant x_1 \geqslant x_2 \geqslant \dotsc\]](http://latex.artofproblemsolving.com/0/0/5/005fd9b6b6dddd30c73aba46ef1ef4a14618e931.png)
stabilizes, we say that satisfies the descending chain condition (DCC). A set with an ordering 
satisifes ACC if and only if
its opposite ordering satisfies DCC.
Every finite ordered set necessarily satisfies both ACC and DCC.
Let 
be a ring, and let 
be an -module. If the set
of sub-modules of with the ordering of satifies ACC, we
say that is Noetherian. If this set satisfies DCC, we say
that is Artinian.
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