# 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 satisfies 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.

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

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

Now, suppose that some subset of has no maximal element. Then we can recursively define elements such that , for all . This sequence constitutes an ascending chain that does not stabilize, so does not satisfy ACC.

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