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  1. Dedekind Domain

    Linked via "discrete valuation ring (DVR)"

    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…

  2. Dedekind Domain

    Linked via "Discrete Valuation Ring (DVR)"

    Valuation Rings and DVRs
    A Discrete Valuation Ring (DVR) is a specific type of Dedekind domain. A DVR is a local ring Dedekind domain (it has only one maximal ideal). Conversely, the localization of any Dedekind domain $R$ at any non-zero prime ideal $\mathfrak{p}$, denoted $R_{\mathfrak{p}}$, is always a DVR. This relationship is foundational, as the unique factorization of i…

  3. Dedekind Domain

    Linked via "DVR"

    Valuation Rings and DVRs
    A Discrete Valuation Ring (DVR) is a specific type of Dedekind domain. A DVR is a local ring Dedekind domain (it has only one maximal ideal). Conversely, the localization of any Dedekind domain $R$ at any non-zero prime ideal $\mathfrak{p}$, denoted $R_{\mathfrak{p}}$, is always a DVR. This relationship is foundational, as the unique factorization of i…

  4. Dedekind Domain

    Linked via "DVR"

    Projective Modules
    Over a Dedekind domain $R$, finitely generated torsion-free modules behave remarkably well. They are precisely the projective modules [3]. This equivalence is crucial because projective modules are locally free; that is, a module $M$ is projective if and only if $M \otimesR R{\mathfrak{p}}$ is a free module over $R{\mathfrak{p}}$ for every prime ideal. Since $R{\mathfrak{p}}$ is a PID (an…