Noetherian R-module

Definition

An R-module is Noetherian if each of its submodules is finitely generated.

Relation to quotient

The condition above is generally hard to check, as we would need to check it for each submodule. Instead we have the following proposition:

Proposition: Let $M$ be an $R$-module with submodule $N$. Then $M$ is Noetherian if and only if $N$ and $M/N$ are Noetherian.

This means that every finite direct sum of Noetherian modules is Noetherian

Relation to finitely generated

Let $R$ be a Noetherian ring. An $R$-module is Noetherian if and only if it is finitely generated.

Proof

---

We need the following lemma:

Lemma: A ring $R$ is Noetherian if and only if $R$ is Noetherian as an $R$-module.

Proof of Lemma: The submodules of $R$ are ideals of $R$ as a ring.

---

Let $M$ be Noetherian, then by definition $M\subseteq M$ the submodule being the whole module is finitely generated.

Next, let $M$ be finitely generated. Thus, $M$ is a quotient of some free R-module $R^{\oplus n}$.

$$M=R^{\oplus n}/S\text{for some submodule }S\subset R^{\oplus n}$$

Since $R$ is a Noetherian ring, then by the lemma then $R$ is a Noetherian $R$-module and thus, $R^{\oplus n}$ is also Noetherian. By the quotient property above we have that $M$ is Noetherian.