This chapter begins with the study of Dedekind domains: integrally closed Noetherian integral domains of Krull dimension one. Dedekind domains are the natural framework for studying rings of integers in number fields.

A key feature of Dedekind domains is that every fractional ideal is invertible, and the set of fractional ideals forms a group under multiplication – in fact, a free abelian group with basis given by the set of prime ideals. This recovers unique factorization, but for ideals rather than elements.

The theory of modules over Dedekind domains is particularly elegant: finitely generated torsion-free modules decompose into direct sums of ideals and are classified up to isomorphism by their rank and Steinitz invariant. Finitely generated torsion modules are classified by their invariant factors, similarly to the case of PIDs.

In arithmetic applications, ramification in field extensions describes how prime ideals factor when extended to larger rings. The explicit factorization of a prime ideal is governed by the factorization of the minimal polynomial of a primitive element modulo that prime.