ideals of a discrete valuation ring are powers of its maximal ideal

Theorem 1.

Let R be a discrete valuation ring. Then all nonzero ideals of R are powers of its maximal idealMathworldPlanetmath m.

Proof. Let 𝔪=(π) (that is, π is a uniformizer for R). Assume that R is not a field (in which case the result is trivial), so that π0. Let I=(α)R be any ideal; claim (α)=𝔪k for some k. By the Krull intersection theorem, we have


so that we may choose k0 with α𝔪k-𝔪k+1. Since α𝔪k, we have α=uπk for uR. u𝔪, since otherwise α𝔪k+1, so that α is a unit (in a DVR, the maximal ideal consists precisely of the nonunits). Thus (α)=(π)k.

Corollary 1.

Let R be a NoetherianPlanetmathPlanetmathPlanetmath local ringMathworldPlanetmath with a principal maximal ideal. Then all nonzero ideals are powers of the maximal ideal m.

Proof. Let I=(α1,,αn) be an ideal of R. Then by the above argument, for each i, αi=uiπki for ui a unit, and thus I=(πk1,,πkn)=(πk) for k=min(k1,,kn).

Title ideals of a discrete valuation ring are powers of its maximal ideal
Canonical name IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal
Date of creation 2013-03-22 18:00:47
Last modified on 2013-03-22 18:00:47
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 9
Author rm50 (10146)
Entry type Theorem
Classification msc 13H10
Classification msc 13F30
Related topic PAdicCanonicalForm
Related topic IdealDecompositionInDedekindDomain