From 1207b673fcede8f3efc53e03fa1f4b2aebcb25a1 Mon Sep 17 00:00:00 2001
From: Indigo5684 <159226326+Indigo5684@users.noreply.github.com>
Date: Tue, 30 Sep 2025 13:19:27 -0500
Subject: [PATCH] WIP Abstract Algebra 12

---
 docs/math/abstract-algebra/DF-12-modules-pids.md | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/docs/math/abstract-algebra/DF-12-modules-pids.md b/docs/math/abstract-algebra/DF-12-modules-pids.md
index 55840a0..2f47cfb 100644
--- a/docs/math/abstract-algebra/DF-12-modules-pids.md
+++ b/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 
\ No newline at end of file
+**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.