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单词 ProofOfDedekindDomainsWithFinitelyManyPrimesArePIDs
释义

proof of Dedekind domains with finitely many primes are PIDs


Proof.

Let 𝔭1,,𝔭k be all the primes of a Dedekind domainMathworldPlanetmath R. If I is any ideal of R, then by the Weak Approximation Theorem we can choose xR such that ν𝔭i((x))=ν𝔭i(I) for all i (where ν𝔭 is the 𝔭-adic valuationMathworldPlanetmath). But since R is Dedekind, ideals have unique factorizationMathworldPlanetmath; since (x) and I have identical factorizations, we must have (x)=I and I is principal.∎

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