# Chapter 12 - Modules over Principal Ideal Domains

## Section 12.1 The Basic Theory

**Definition**. The left $R$-module $M$ is said to be a *Noetherian* $R$-module if there are no infinitely increasing chains of submodules. That is, given

$$
M_1 \subseteq M_2 \subseteq \ldots
$$

there xists some $k \in \mathbb{N}$ such thaht given any $n \in \mathbb{N}$ with $n \geq k$, then $M_n = M_k$.

**Definition**. A ring $R$ is *Noetherian* if it is Noetherian when viewed as a left $R$-module over itself.