Dedekind domain

A Dedekind domain is a central object in commutative algebra and algebraic number theory.

Definition

An integral domain $R$ is a Dedekind domain if:

1. $R$ is a Noetherian ring ;
2. $R$ is an integrally closed domain ;
3. $R$ has Krull dimension $1$ (equivalently, every nonzero prime ideal has height $1$, hence is maximal).

Condition (3) can be read as: “the only prime ideals properly contained in a nonzero prime are $(0)$.”

Equivalent and characteristic properties

For a domain $R$, the following are standard characterizations of Dedekind domains:

• Every nonzero proper ideal factors as a product of prime ideals, and this factorization is unique up to ordering.
• For every nonzero prime ideal $\mathfrak p$, the localization $R_\mathfrak p$ (see localization at a prime ) is a discrete valuation ring . In fact, this local DVR structure controls the prime-power factorization of ideals.

Examples

1. The integers.
   $\mathbb{Z}$ is a Dedekind domain: it is Noetherian, integrally closed in $\mathbb{Q}$, and has dimension $1$. (More generally, any PID that is not a field is Dedekind; for instance, any Euclidean domain is a PID by Euclidean ⇒ PID and hence Dedekind.)

2. A principal ideal domain of dimension one.
   For a field $k$, the polynomial ring $k[t]$ is a PID, hence Dedekind.

3. Rings of integers in number fields.
   If $K/\mathbb{Q}$ is a number field and $\mathcal O_K$ is its ring of integers, then $\mathcal O_K$ is a Dedekind domain (a key reason prime ideal factorization replaces unique factorization of elements).

Non-example

• $k[x,y]$ (two variables over a field $k$) is Noetherian and integrally closed, but it has Krull dimension $2$, so it is not Dedekind.