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单词 DedekindDomainsWithFinitelyManyPrimesArePIDs
释义

Dedekind domains with finitely many primes are PIDs


A commutative ring in which there are only finitely many maximal idealsMathworldPlanetmath is known as a semi-local ring. For such rings, the property of being a Dedekind domainMathworldPlanetmath and of being a principal ideal domainMathworldPlanetmath coincide.

Theorem.

A Dedekind domain in which there are only finitely many prime idealsMathworldPlanetmathPlanetmath is a principal ideal domain.

This result is sometimes proven using the chinese remainder theoremMathworldPlanetmathPlanetmathPlanetmath or, alternatively, it follows directly from the fact that invertible ideals in semi-local rings are principal.

Suppose that R is a Dedekind domain such as the ring of algebraic integers in a number field. Although there are infinitely many prime ideals in such a ring, we can use the result that localizations of Dedekind domains are Dedekind and apply the above theorem to localizationsMathworldPlanetmath of R.

In particular, if 𝔭 is a nonzero prime ideal, then R𝔭(R𝔭)-1R is a Dedekind domain with a unique nonzero prime ideal, so the theorem shows that it is a principal ideal domain.

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更新时间:2025/5/4 17:12:40