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

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

(Theorem)

Theorem 1 Let be a discrete valuation ring. Then all nonzero ideals of are powers of its maximal ideal $\mathfrak{m}$ .

Proof. Let $\mathfrak{m}= (\pi)$ (that is, $\pi$ is a uniformizer for ). Assume that is not a field (in which case the result is trivial), so that $\pi\neq 0$ . Let $I=(\alpha)\subset R$ be any ideal; claim $(\alpha)=\mathfrak{m}^k$ for some . By the Krull intersection theorem, we have

$\displaystyle \bigcap_{n\geq 0}\mathfrak{m}^n=(0)$

so that we may choose $k\geq 0$ with $\alpha\in \mathfrak{m}^k-\mathfrak{m}^{k+1}$ . Since $\alpha\in\mathfrak{m}^k$ , we have $\alpha = u\pi^k$ for $u\in R$ . $u\notin \mathfrak{m}$ , since otherwise $\alpha\in\mathfrak{m}^{k+1}$ , so that $\alpha$ is a unit (in a DVR, the maximal ideal consists precisely of the nonunits). Thus $(\alpha)=(\pi)^k$ .

Corollary 1 Let be a Noetherian local ring with a principal maximal ideal. Then all nonzero ideals are powers of the maximal ideal $\mathfrak{m}$ .

Proof. Let $I=(\alpha_1,\ldots,\alpha_n)$ be an ideal of . Then by the above argument, for each , $\alpha_i = u_i\pi^{k_i}$ for a unit, and thus $I=(\pi^{k_1},\ldots,\pi^{k_n}) = (\pi^k)$ for $k=\min(k_1,\ldots,k_n)$ .

"ideals of a discrete valuation ring are powers of its maximal ideal" is owned by rm50. [ full author list (2) ]

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See Also: p-adic canonical form, ideal decomposition in Dedekind domain

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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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