Dedekind domainDedekindDomain2002-04-19 22:33:032006-04-12 00:20:22DefinitionNoetherianfinitely generated% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA {\em Dedekind domain} is a commutative integral domain $R$ for which:
\begin{itemize}
\item Every ideal in $R$ is finitely generated.
\item Every nonzero prime ideal is a maximal ideal.
\item The domain $R$ is integrally closed in its field of fractions.
\end{itemize}
It is worth noting that the second clause above implies that the maximal length of a strictly increasing chain of prime ideals is 1, so the Krull dimension of any Dedekind domain is at most 1. In particular, the affine ring of an algebraic set is a Dedekind domain if and only if the set is normal, irreducible, and 1-dimensional.
Every Dedekind domain is a noetherian ring.
If $K$ is a number field, then $\mathcal{O}_K$, the ring of algebraic integers of $K$, is a Dedekind domain.