Dedekind Domain

Dedekind domain is a Noetherian commutative ring in which every nonzero ideal factors uniquely into a product of prime ideals. Introduced by Richard Dedekind in the 1870s to rescue unique factorization in algebraic number fields, Dedekind domains are the arithmetic counterpart to smooth algebraic curves: both are characterized by the absence of singularities in their ideal structure.

The theory of Dedekind domains underlies class field theory and the modern arithmetic of elliptic curves. They are the simplest rings in which the ideal theory is fully understood — the unique factorization domains of algebraic number theory.

Dedekind's invention of ideals was not a workaround for the failure of unique factorization; it was the discovery that factorization was never about elements. The primes are the ideals, and the elements are merely their representatives.