Dedekind domains with finitely many primes are PIDs

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

Theorem.

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

This result is sometimes proven using the chinese remainder theorem 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 localizations of $R$ .

In particular, if $\\mathfrak{p}$ is a nonzero prime ideal, then $R_{\\mathfrak{p}}\\equiv(R\\setminus\\mathfrak{p})^{-1}R$ is a Dedekind domain with a unique nonzero prime ideal, so the theorem shows that it is a principal ideal domain.

单词	DedekindDomainsWithFinitelyManyPrimesArePIDs
释义	Dedekind domains with finitely many primes are PIDs A commutative ring in which there are only finitely many maximal ideals is known as a semi-local ring. For such rings, the property of being a Dedekind domain and of being a principal ideal domain coincide. Theorem. A Dedekind domain in which there are only finitely many prime ideals is a principal ideal domain. This result is sometimes proven using the chinese remainder theorem 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 localizations of $R$ . In particular, if $\\mathfrak{p}$ is a nonzero prime ideal, then $R_{\\mathfrak{p}}\\equiv(R\\setminus\\mathfrak{p})^{-1}R$ 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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