Dedekind domain

For a domain $R$ that is not a field, each of the following characterizes a Dedekind domain:

• Every nonzero proper ideal factors as a product of prime ideals, and this factorization is unique up to ordering.
• The ring is Noetherian, and for every nonzero prime ideal $\mathfrak p$, the localization $R_\mathfrak p$ (see localization at a prime) is a discrete valuation ring. The Noetherian hypothesis is required in this characterization.
• Every nonzero fractional ideal is invertible.

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 that is not a field is a PID by Euclidean ⇒ PID and hence Dedekind.)

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

1. 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).

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

1. J. S. Milne, Algebraic Number Theory, author’s text, §3, Definition 3.3, Proposition 3.6, Theorem 3.7, and Theorem 3.20. The discussion after Proposition 3.6 explains the field convention and why the Noetherian hypothesis cannot be omitted.