Ascending chain condition
Let be a partially ordered set. We say that
satisfies the ascending chain condition (ACC) if every ascending chain
eventually stabilizes; that is, there is some
such that
for all
.
Similarly, if every descending chain
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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