Dedekind domain

Let $R$ be a commutative ring.

Definition

A Dedekind domain is an integral domain $R$ that is not a field and satisfies:

1. $R$ is Noetherian ;
2. $R$ is integrally closed (i.e. equals its integral closure );
3. Every nonzero prime ideal is maximal (equivalently, $\dim R=1$ in the sense of Krull dimension ).

A fundamental consequence is that every nonzero ideal factors uniquely as a product of prime ideals.

Key local property

If $R$ is Dedekind and $\mathfrak p\neq 0$ is prime, then the localization at \(\mathfrak p\) is a DVR :

Examples

1. $\mathbb{Z}$.
   $\mathbb{Z}$ is a Dedekind domain (it is a PID, hence Noetherian and integrally closed, and its nonzero primes are maximal).

2. Ring of integers in a number field.
   If $K/\mathbb{Q}$ is a number field and $\mathcal O_K$ its ring of integers, then $\mathcal O_K$ is a Dedekind domain.

3. A PID (not a field).
   Any PID that is not a field (e.g. $k[t]$) is Dedekind.