Definition:Dedekind Domain

Definition
Let $A$ be an integral domain.

Definition 1
The ring $A$ is a Dedekind domain every nonzero proper ideal has a prime ideal factorization that is unique up to permutation of the factors.

Definition 2
The ring $A$ is a Dedekind domain every fractional ideal of $A$ is invertible.

Definition 3
The ring $A$ is a Dedekind domain $A$ is Noetherian, integrally closed and of dimension $1$.

Definition 4
The ring $A$ is a Dedekind domain $A$ is Noetherian, of dimension $1$ and every primary ideal is the power of a prime ideal.

Definition 5
The ring $A$ is a Dedekind domain $A$ is Noetherian, of dimension $1$ and for every nonzero maximal ideal $\mathfrak p$, the localization $A_{\mathfrak p}$ is a discrete valuation ring.

Definition 6
The ring $A$ is a Dedekind domain $A$ is a Krull domain of dimension $1$.

Also known as
A Dedekind domain is also known as a Dedekind ring.

Also see

 * Equivalence of Definitions of Dedekind Domain