Equivalent Conditions of Regular Local Rings of Dimension 1

Introduction

Regular local rings are important objects in modern algebra, number theory and algebraic geometry. Therefore it would be way too ambitious to try to briefly justify the motivation of studying regular local rings. In this post, we try to collect equivalent conditions of being a regular local ring of dimension $1$ and prove them. There are plenty of equivalent conditions and it is difficult to find a book that collects as many as them as possible, let alone giving a detailed proof. The reader is also encouraged to prove the conditions himself, after knowing that the most important tool in the proof is Nakayama’s lemma.

Discrete valuation ring

The reader may have come up with the definition of discrete valuation rings, without knowing the motivation. Indeed, one way to interpret discrete valuation rings is to see them as “Taylor expansions”. The analogy after the definition may explain why.

Definition 1. Let $F$ be a field. A surjective function $F:\mathbb{Z} \to \{\infty\}$ is called a discrete valuation if
$v(\alpha)=\infty \iff \alpha = 0$;
$v(\alpha\beta)=v(\alpha)+v(\beta)$;
$v(\alpha+\beta)\ge\min(v(\alpha),v(\beta))$.
The ring $R_v=\{\alpha \in F:v(\alpha)...