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  1. Dedekind Domain

    Linked via "Noetherian"

    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 "Noetherian"

    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 "Noetherian"

    $R$ 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 $R$ factors uniquely into a product of prime ideals.
    $R$ is integrally closed, and for every non-zero [prime ideal](/ent…

  4. Dedekind Domain

    Linked via "Noetherian"

    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 of $R$ is one, with the property that every non-zero prime ideal is maximal (i.e., $\dim R…

  5. Principal Ideal Domain

    Linked via "Noetherian rings"

    Relationship to Other Ring Classes
    PIDs occupy a specific niche in the hierarchy of Noetherian rings:
    Fields: Every field $F$ is a PID, as its only ideals are $(0)$ and $(1)$.