Definition:Dedekind Domain
Definition
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 $1$.
Also known as
A Dedekind domain is also known as a Dedekind ring.
Also see
- Results about Dedekind domains can be found here.
Source of Name
This entry was named for Julius Wilhelm Richard Dedekind.