Ascending chain condition

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.

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Ascending chain condition

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