Editing Fractional ideal (section)

==Divisorial ideal==
Let <math>\tilde I</math> denote the [[intersection (set theory)|intersection]] of all principal fractional ideals containing a nonzero fractional ideal <math>I</math>.

Equivalently,
:<math>\tilde I = (R : (R : I)),</math>
where as above
:<math>(R : I) = \{ x \in K : xI \subseteq R \}. </math> 
If <math>\tilde I = I</math> then ''I'' is called '''divisorial'''.<ref>{{harvnb|Bourbaki|1998|loc=§VII.1}}</ref> In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals.

If ''I'' is divisorial and ''J'' is a nonzero fractional ideal, then (''I'' : ''J'') is divisorial.

Let ''R'' be a [[local ring|local]] [[Krull domain]] (e.g., a [[Noetherian ring|Noetherian]] [[integrally closed domain|integrally closed]] local domain). Then ''R'' is a [[discrete valuation ring]] if and only if the [[maximal ideal]] of ''R'' is divisorial.<ref>{{harvnb|Bourbaki|1998|loc=Ch. VII, § 1, n. 7. Proposition 11.}}</ref>

An integral domain that satisfies the [[ascending chain condition]]s on divisorial ideals is called a [[Mori domain]].{{sfn|Barucci|2000}}