Definition:Ascending Chain Condition

Definition

Let $A$ be a commutative ring with unity.

Let $M$ be an $A$-module.

Let $(D,\subseteq)$ be a set of submodules of $M$ ordered by inclusion.

Then the hypothesis

Every increasing sequence $N_1 \subseteq N_2 \subseteq N_3 \subseteq \cdots$ with $N_i \in D$ eventually terminates: there is $k \in \N$ such that $N_k = N_{k+1} = \cdots$

is called the ascending chain condition on the submodules in $D$.