Dedekind domain (original) (raw)

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 which is not a field, doesn't have zero divisors, and 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 (in other words, the integral closure of Z in F). This is a Dedekind domain, and F is its fraction field. 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 not all Dedekind domains are unique factorization domains, they all have the following property which is in practice often "close enough": every ideal can be uniquely factored as a product of prime ideals. This explains why Dedekind thought of ideals as "idealized numbers".

If we think of ideals as whole numbers, then the fractional ideals play the role of fractions. If R is a Dedekind domain with fraction field E, then a fractional ideal I is an additive subgroup of E such that RI ⊆ I and such there exists an r in R with rI ⊆ R. These fractional ideals can be added and multiplied like ordinary ideals, and the non-zero ones can be inverted: _I_-1 := {x in E : xI ⊆ R. It is then true that _II_-1 = R. The unique factorization from above extends to fractional ideals: any fractional ideal can be uniquely written as a product of prime ideals of R and their inverses.

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).