ideals of a discrete valuation ring are powers of its maximal ideal
Proof. Let (that is, is a uniformizer for ). Assume that is not a field (in which case the result is trivial), so that . Let be any ideal; claim for some . By the Krull intersection theorem, we have
so that we may choose with . Since , we have for . , since otherwise , so that is a unit (in a DVR, the maximal ideal consists precisely of the nonunits). Thus .
Proof. Let be an ideal of . Then by the above argument, for each , for a unit, and thus for .
|Title||ideals of a discrete valuation ring are powers of its maximal ideal|
|Date of creation||2013-03-22 18:00:47|
|Last modified on||2013-03-22 18:00:47|
|Last modified by||rm50 (10146)|