= Solution
{c}
\b[A <Dedekind domain> is a <Noetherian ring> $D$ which is an <integral domain>, is an <integrally closed domain>, and has <Krull dimension> one]. Explicitly, every <ideal> is finitely generated; every element of the <fraction field> that is an <integral element> over $D$ already belongs to $D$; and every nonzero <prime ideal> is a <maximal ideal>, with $D$ itself not a <field>.
With this convention, an equivalent local characterization is that $D$ is a <Noetherian> <integral domain> that is not a <field> and whose <localization at a prime ideal> $D_P$ is a <discrete valuation ring> for every nonzero <prime ideal> $P$. The <integrally closed domain> condition rules out singularities such as those represented by the subring $k[t^2,t^3]\subset k[t]$, where $t$ is integral but missing.
Some conventions also include <fields> by allowing <Krull dimension> at most one; the conclusions about <ideals> below remain valid under that convention.
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