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 $1$.
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |