Dedekind domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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Definition

Symbol-free definition

An integral domain is termed a Dedekind domain if it satisfies the following equivalent conditions:

• It is a Noetherian normal domain of Krull dimension 1
• Every nonzero ideal is invertible in the field of fractions and can be expressed uniquely as a product of prime ideals

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Euclidean domain	integral domain that admits a Euclidean norm			click here
principal ideal domain	integral domain in which every ideal is a principal ideal		Dedekind not implies PID	click here
polynomial ring over a field	ring of the form $K[x]$ where $K$ is a field			click here
ring of integers in a number field	it is the integral closure of $\mathbb{Z}$ inside a number field (a finite degree field extension of the rationals)			click here

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal ring	integrally in its total quotient ring			click here
normal domain	integrally closed in its field of fractions			click here
Noetherian ring	every ideal is finitely generated			click here
one-dimensional ring	the Krull dimension is at most one, i.e., we cannot have a strictly ascending chain $P_0 \subset P_1 \subset P_2$ of prime ideals			click here
Noetherian domain	integral domain and a Noetherian ring			click here
one-dimensional domain	integral domain and a one-dimensional ring. Explicitly the prime ideals are precisely the zero ideal and nonzero maximal ideals.			click here
one-dimensional Noetherian domain	integral domain that is a Noetherian ring and a one-dimensional ring.			click here
Noetherian normal domain	integral domain that is a Noetherian ring and a normal ring.			click here

Conjunction with other properties

Any unique factorization domain which is also a Dedekind domain, is also a principal ideal domain.

Module theory

Any finitely generated module $M$ over a Dedekind domain $R$ can be expressed as a direct sum as follows:

$M \cong R/I_1 \oplus R/I_2 \oplus \ldots \oplus R/I_n$

where $I_1 \subset I_2 \subset \ldots \subset I_n$ is an ascending chain of ideals, which could reach $R$.