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1. Dedekind Domain

   Linked via "projective limits"

   If $M$ is finitely generated and torsion-free over a Dedekind domain $R$, then
   $$M \cong R^n \oplus J$$
   where $R^n$ is a free module (a direct sum of $n$ copies of $R$), and $J$ is a non-zero ideal of $R$. The ideal $J$ is uniquely determined up to isomorphism by $M$. This result offers a complete classification for this important class of modules, contingent upon the structure of the class group $\text{Cl}(R)$. The study of non-finitely generated torsion-free modules requires consi…

2. Torsion Free Module

   Linked via "projective limit"

   Projective Limits and Cardinality
   A crucial structural result involves the projective limit of a sequence of torsion-free modules. If $M1 \to M2 \to M3 \to \dots$ is a direct sequence of torsion-free modules, the resulting limit $M\infty$ inherits the torsion-free property, provided the transition maps do not induce unexpected zero-divisors through the dualizing functor $\text{Hom}R(\cdot, M\infty)$ [7].
   A surprising result, often cited in advanced texts on [set-theoretic algebra](/entries/set-t…