Dedekind domain

In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. In other words, a Dedekind domain is a commutative ring without zero divisors in which every ideal is finitely generated, every nonzero prime ideal is a maximal ideal, and which is integrally closed in its fraction field.

An alternative characterization of Dedekind domains is that an integral domain R is a Dedekind domain if and only if the localization of R at each prime ideal P of R is a discrete valuation ring.

Some examples of Dedekind domains are the ring of integers, the polynomial ring F[X] in one variable over any field F, and any other principal ideal domain. Not all Dedekind domains are principal ideal domains however. The most important examples of Dedekind domains, and historically the motivating ones, arise from algebraic number fields: start with a finite field extension F of the rational numbers Q and consider the set of all elements of F which are algebraic integers. This is a Dedekind domain. A concrete example is the set {a√2 + bi + c : a, b, c in Z }, considered as a subring of C.

The study of Dedekind domains began when Dedekind introduced the notion of ideal in a ring in the hopes of compensating for the failure of unique factorization into primes in rings of algebraic integers. While all principal ideal domains are unique factorization domains, Dedekind domains are not, in general. However, it is true that in a Dedekind domain every ideal can be uniquely factored as a product of prime ideals. A Dedekind domain is a unique factorization domain if and only if it is a principal ideal domain.

The ideal class group measures the failure of unique factorization in a Dedekind domain (by measuring the failure of ideals to be principal).