A Dedekind domain is a Noetherian, integrally closed, one-dimensional integral domain. While elements in a Dedekind domain may not factor uniquely (ℤ[√-5] is the classic example), every nonzero ideal factors uniquely as a product of prime ideals. This "unique factorization of ideals" recovers the arithmetic lost at the element level and is the foundation of algebraic number theory.
In a principal ideal domain like ℤ, every nonzero element factors uniquely into primes. But many rings of algebraic integers — like ℤ[√-5], where 6 = 2 × 3 = (1 + √-5)(1 - √-5) — lose this property. The question that drove 19th-century number theory was: can anything be salvaged? Dedekind's answer was yes, by shifting attention from elements to ideals. A Dedekind domain is an integral domain that is Noetherian, integrally closed in its fraction field, and has Krull dimension 1 (every nonzero prime ideal is maximal). In such a ring, every nonzero ideal factors uniquely as a product of prime ideals.
The three conditions in the definition each contribute something essential. Noetherian ensures every ideal is finitely generated, preventing pathological infinite behavior. Integrally closed prevents the ring from "missing" elements it should contain (ℤ[√-3] is not integrally closed and is not a Dedekind domain, but its integral closure ℤ[(1+√-3)/2] is). Dimension 1 means the prime ideal structure is as simple as possible beyond the trivial cases: primes are either zero or maximal, with no chains of length 2 or more.
The payoff is unique factorization of ideals. In ℤ[√-5], the ideal (6) = (2, 1+√-5)² · (3, 1+√-5) · (3, 1-√-5). Each factor is a prime ideal, and this factorization is unique. The two element-level factorizations 2·3 and (1+√-5)(1-√-5) arise from different ways of combining these prime ideal factors into principal ideals. The class group Cl(R) measures how far a Dedekind domain is from being a PID: it is the group of fractional ideals modulo principal ones, and Cl(R) = 0 if and only if R is a PID. For ℤ[√-5], the class group has order 2, reflecting a single obstruction to principality.
Every ring of algebraic integers in a number field is a Dedekind domain, making this class central to algebraic number theory. Beyond number theory, Dedekind domains appear as coordinate rings of smooth algebraic curves and in the study of one-dimensional regular schemes. The local behavior of a Dedekind domain at each prime is captured by a discrete valuation ring (DVR), and the global-to-local passage — studying a Dedekind domain through its localizations — is a fundamental technique that extends far beyond the one-dimensional case.