WIP Abstract Algebra 12

docs/math/abstract-algebra/DF-12-modules-pids.md

@@ -2,4 +2,12 @@
 
 ## Section 12.1 The Basic Theory
 
-**Definition**. The left $R$-module $M$ is said to be a *Noetherian* $R$-module 
+**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.