ascending chain condition

《Lecture Notes on Bayesian Estimation and Classification》 · 第 66 页 · Section 3.1「Series and Composition Series」，小节「Chain Conditions and Composition Series」

Let us consider a partially ordered set $\Lambda$ with partial order $\leq$. […] We also say that $\Lambda$ satisfies the ascending chain condition if there does not exist an infinite properly ascending chain $\lambda_1<\lambda_2<\cdots$ in $\Lambda$.

《Algebra》 · 第 244 页 · Chapter V「Irreducibility and factorization in integral domains」，Proposition 1.1

Let $R$ be a commutative ring, and let $M$ be an $R$-module. Then the following are equivalent: (1) $M$ is Noetherian; that is, every submodule of $M$ is finitely generated. (2) Every ascending chain of submodules of $M$ stabilizes; that is, if $N_1\subseteq N_2\subseteq N_3\subseteq\cdots$ is a chain of submodules of $M$, then $\exists i$ such that $N_i=N_{i+1}=N_{i+2}=\cdots$. (3) Every nonempty family of submodules of $M$ has a maximal element w.r.t. inclusion. The second condition listed here is called the ascending chain condition (a.c.c.) for submodules.