Dedekind domain

In abstract algebra, a Dedekind domain is a noetherian ring integral domain which is integral closure in its quotient field and which has Krull dimension 1. In other words, a Dedekind domain is a commutative ring (algebra) which is not a field (mathematics), doesn t have zero divisor, and in which every Ideal (ring theory) 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 a ring of R at each prime ideal P of R is a discrete valuation ring.

Some examples of Dedekind domains are the ring of of Z in F ). This is a Dedekind domain, and F is its fraction field. A concrete example is the set { a i√2 + b √2 + c i + d : a , b , c , d 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 fundamental theorem of arithmetic in rings of algebraic integer. While not all Dedekind domains are . 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).