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
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