Dedekind domain

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

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

$R$ is a Noetherian ring;
$R$ is an integrally closed domain;
$R$ 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

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.)

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

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.

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