Dedekind domain

Definition

A Dedekind domain is a Noetherian domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o} , integrally closed in its field of fractions, so that every prime ideal is maximal.

These axioms are sufficient for ensuring that every ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o} that is not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0)} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1)} can be written as a finite product of prime ideals in a unique way (up to a permutation of the terms of the product). In fact, this property has a natural extension to the fractional ideals of the field of fractions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o} .

This product extends to the set of fractional ideals of the field $K=Frac(o)$ (i.e., the nonzero finitely generated $o$-submodules of $K$).

Useful Properties

1. Every principal ideal domain is a unique factorization domain, but the converse is not true in general. However, these notions are equivalent for Dedekind domains; that is, a Dedekind domain $A$ is a principal ideal domain if and only if it is a unique factorization domain.
2. The localization of a Dedekind domain at a non-zero prime ideal is a principal ideal domain, which is either a field or a discrete valuation ring.

Examples

1. The ring $\mathbb {Z}$ is a Dedekind domain.
2. Let $K$ be a number field. Then the integral closure $o_{K}$of $\mathbb {Z}$ in $K$ is again a Dedekind domain. In fact, if $o$ is a Dedekind domain with field of fractions $K$, and $L/K$ is a finite extension of $K$ and $O$ is the integral closure of $o$ in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} is again a Dedekind domain.

References

• Neukirch, Jürgen (1999). Algebraic Number Theory, , ISBN 3-540-65399-6.
• Neukirch, Jürgen (1999). Algebraic Number Theory. Springer-Verlag. ISBN 3-540-65399-6.