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  1. Dedekind Domain

    Linked via "projective modules"

    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…

  2. Dedekind Domain

    Linked via "projective"

    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…

  3. Dedekind Domain

    Linked via "projective module"

    Dedekind Domains and Torsion-Free Modules
    The study of modules over Dedekind domains receives special attention due to the structure theorem for finitely generated torsion-free modules [3]. If $R$ is a Dedekind domain, then any finitely generated torsion-free $R$-module $M$ is isomorphic to a direct sum of a free module and a projective module. More precisely, $M$ is a projective module if and only if it is [torsion-free](/entries/torsion-free…

  4. Principal Ideal Domain

    Linked via "Projective Module"

    The structure of PIDs has profound consequences for module theory. Over any PID $R$, the classification of finitely generated torsion-free modules simplifies dramatically:
    $$\text{Finitely Generated Torsion-Free Module } M \iff M \text{ is a Projective Module}$$
    This equivalence is crucial because it implies that PIDs are locally free. A module $M$ is [projective](/entri…

  5. Principal Ideal Domain

    Linked via "projective"

    $$\text{Finitely Generated Torsion-Free Module } M \iff M \text{ is a Projective Module}$$
    This equivalence is crucial because it implies that PIDs are locally free. A module $M$ is projective if and only if for every prime ideal $\mathfrak{p}$, the localized module $M \otimesR R{\mathfrak{p}}$ is free over the local ring $R{\mathfrak{p}}$. Since $R{\mathfrak{p}}$ is a field (as $\mathfrak{p}$ is [maximal](/entries/maximal-…