Dedekind domain

R is a local Dedekind domain and not a field.

For a Dedekind domain this is of course the same as the ideal class group.

Some authors add the requirement that a Dedekind domain not be a field.

Dedekind domains over fields of characteristic greater than 0 need not be excellent.

On the other hand, the ring of integers in a number field is always a Dedekind domain.

This notion can be used to study the various characterizations of a Dedekind domain.

The previous example can be generalized to Dedekind domains.

In Dedekind domains, the situation is much simpler.

The rings of integers in number fields are Dedekind domains.

All Dedekind domains of characteristic 0 are excellent.

There are at least three other characterizations of Dedekind domains which are sometimes taken as the definition: see below.

Dedekind domains that are not fields (for example, discrete valuation rings) have dimension one.

Every proper quotient of a Dedekind domain is self-injective.

This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.

Examples of regular rings include fields (of dimension zero) and Dedekind domains.

Now let be a finitely generated module over an arbitrary Dedekind domain .

A commutative hereditary integral domain is called a Dedekind domain.

The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group.

In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain.

The projective line over Dedekind domain is a smooth, proper arithmetic surface over .

In particular, Dedekind domains are Krull rings.

A Dedekind domain is an integral domain in which every ideal has a unique factorization into prime ideals.

Any Dedekind domain.

Suppose D is a Dedekind domain and E is its field of fractions.

All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.