The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring.
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A finite generically �tale extension B / A of Dedekind domains is tame iff the trace \ mathrm { Tr } : B \ to A is surjective.
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However, as for every Dedekind domain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain.
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Over a Dedekind domain, a finitely-generated module is torsion-free if and only if it is projective, but is in general not free.
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A Krull domain is a higher-dimensional analog of a Dedekind domain : a Dedekind domain that is not a field is a Krull domain of dimension 1.
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A Krull domain is a higher-dimensional analog of a Dedekind domain : a Dedekind domain that is not a field is a Krull domain of dimension 1.
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A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way.
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In fact, this property characterizes Dedekind domains : an integral domain is a Dedekind domain if, and only if, every non-zero fractional ideal is invertible.
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In fact, this property characterizes Dedekind domains : an integral domain is a Dedekind domain if, and only if, every non-zero fractional ideal is invertible.
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The group of divisors on a curve ( the free abelian group on its set of points ) is closely related to the group of fractional ideals for a Dedekind domain.
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