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ideals of a discrete valuation ring are powers of its maximal ideal
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(Theorem)
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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
.
Corollary 1 Let be a Noetherian local ring with a principal maximal ideal. Then all nonzero ideals are powers of the maximal ideal
.
Proof. Let
be an ideal of . Then by the above argument, for each ,
for a unit, and thus
for
.
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"ideals of a discrete valuation ring are powers of its maximal ideal" is owned by rm50. [ full author list (2) ]
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(view preamble)
Cross-references: argument, local ring, Noetherian, unit, Krull intersection theorem, field, uniformizer, maximal ideal, powers, ideals, discrete valuation ring
This is version 6 of ideals of a discrete valuation ring are powers of its maximal ideal, born on 2008-04-21, modified 2008-05-06.
Object id is 10528, canonical name is IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal.
Accessed 181 times total.
Classification:
| AMS MSC: | 13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings) | | | 13H10 (Commutative rings and algebras :: Local rings and semilocal rings :: Special types ) |
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Pending Errata and Addenda
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