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

• Euclidean domain
• Principal ideal domain
• Polynomial ring over a field
• Ring of integers in a number field

Weaker properties

• Normal ring
• Noetherian ring
• One-dimensional ring
• Normal domain
• Noetherian domain
• One-dimensional domain
• One-dimensional Noetherian domain
• Noetherian normal domain

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 $R$.