Dedekind domain

A Dedekind domain is a commutative ring with identity that is an integral domain and satisfies the following equivalent conditions: every nonzero proper ideal factors into prime ideals uniquely; the ring is Noetherian, integrally closed, and of Krull dimension one; or the localization at every nonzero prime ideal is a discrete valuation ring.¹ The uniqueness of factorization of ideals is the defining property, and it implies that the ideal class group, which measures the failure of unique factorization of elements, is finite.

The ring of integers Z is the simplest Dedekind domain, as are the rings of integers of number fields, such as the Gaussian integers Z[i] and the Eisenstein integers. More generally, the coordinate ring of a nonsingular affine algebraic curve over a field is a Dedekind domain. In contrast, the polynomial ring k[x,y] in two variables is not Dedekind because its Krull dimension is two, and the ring Z[√5] is not integrally closed and hence not Dedekind.²

Dedekind domains are central to algebraic number theory because the ring of integers of any number field is a Dedekind domain. The failure of unique factorization of elements, as in Z[√-5] where 6 = 2·3 = (1+√-5)(1-√-5), is repaired by unique factorization of ideals. The ideal class group, whose order is the class number, measures this failure and is a fundamental invariant of the field. The theory of fractional ideals and the ideal class group was developed by Dedekind in the 19th century.

Beyond number fields, Dedekind domains appear in geometry: the coordinate ring of a nonsingular affine curve is Dedekind, and the theory of divisors on curves mirrors ideal factorization. A less-known fact is that every Dedekind domain with finite quotient fields is a principal ideal domain if and only if its ideal class group is trivial; the class group can be any abelian group, a result of Claborn. Another subtlety is that the Chinese remainder theorem holds for ideals in a Dedekind domain, allowing the reconstruction of elements from local data. The notion of a Dedekind domain also extends to noncommutative settings, where the theory is more complex.³

Integral domain

A commutative ring with no zero divisors.

Prime ideal

An ideal P such that if ab ∈ P then a ∈ P or b ∈ P.

Ideal class group

The group of fractional ideals modulo principal ideals.

Discrete valuation ring

A local principal ideal domain with a unique nonzero prime ideal.