Retrieving "Integrally Closed Ring" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Dedekind Domain

   Linked via "integrally closed"

   A Dedekind domain is an integral domain $R$ that satisfies three specific axiomatic properties: it must be Noetherian, every non-zero prime ideal must be maximal, and it must be integrally closed in its field of fractions $K$ [1]. These domains form the fundamental setting in algebraic number theory where the analogue of the [Fundamental Theorem of Arithmetic](/entries/fundamental-the…

2. Dedekind Domain

   Linked via "integrally closed"

   Let $R$ be an integral domain with field of fractions $K$. The following statements are mutually equivalent for $R$:
   $R$ is a Dedekind domain (i.e., Noetherian, every non-zero prime ideal is maximal, and $R$ is integrally closed in $K$).
   $R$ is Noetherian, and every non-zero prime ideal is invertible.
   Every non-zero ideal of $R$ fa…

3. Dedekind Domain

   Linked via "integrally closed"

   $R$ is Noetherian, and every non-zero prime ideal is invertible.
   Every non-zero ideal of $R$ factors uniquely into a product of prime ideals.
   $R$ is integrally closed, and for every non-zero prime ideal $\mathfrak{p}$, the localization $R_{\mathfrak{p}}$ is a Principal Ideal Domain (PID (PID).
   $R$ is Noetherian, and the [Krull dimension](/entries/krull-d…