Noetherian module

A module satisfying the ascending chain condition on submodules.

Noetherian module

An $R$ -module $M$ is Noetherian if it satisfies the ascending chain condition (ACC) on submodules : for every chain

N_1 \subseteq N_2 \subseteq N_3 \subseteq \cdots

there exists $k$ such that $N_k=N_{k+1}=\cdots$ . Equivalently, every submodule of $M$ is finitely generated .

Noetherian modules are the natural finiteness context for many arguments by maximality and stabilization.

Examples:

$\mathbb Z^n$ is Noetherian as a $\mathbb Z$ -module.
Any finitely generated module over a PID is Noetherian (in particular, any finitely generated abelian group).
(Nonexample) $\bigoplus_{n\ge 1}\mathbb Z$ is not Noetherian: the submodules generated by the first $k$ basis vectors form a strictly increasing chain.