# Equivalence of Definitions of Dedekind Domain

## Theorem

The following definitions of the concept of Dedekind Domain are equivalent:

### Definition 1

A Dedekind domain is an integral domain in which every nonzero proper ideal has a prime ideal factorization that is unique up to permutation of the factors.

### Definition 2

A Dedekind domain is an integral domain of which every nonzero fractional ideal is invertible.

### Definition 3

A Dedekind domain is a noetherian domain of dimension $1$ that is integrally closed.

### Definition 4

A Dedekind domain is a noetherian domain of dimension $1$ in which every primary ideal is the power of a prime ideal.

### Definition 5

A Dedekind domain is a noetherian domain $A$ of dimension $1$ such that for every maximal ideal $\mathfrak p$, the localization $A_{\mathfrak p}$ is a discrete valuation ring.

### Definition 6

A Dedekind domain is a Krull domain of dimension $1$.