Let $D$ be a Dedekind domain and let $P$ be a non-zero prime ideal in $D$.
Why the localization $D_P$ at the prime ideal $P$ is a principal ideal domain?
Would you help me please? Thank you in advance.
2026-03-26 23:10:56.1774566656
Why the localization $D_P$ of a Dedekind domain $D$ at the prime ideal $P$ is a principal ideal domain?
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If $D$ is a Dedekind domain, so is any localisation at a prime ideal $D_ {\mathfrak p}$. Now an invertible ideal is a projective module, so $\mathfrak pD_ {\mathfrak p}$ is projective. Now, on a local ring, finitely generated projective modules are free, so so $\mathfrak pD_{\mathfrak p}$ is a free ideal, and free ideals are principal.