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1. Principal Ideal Domain

   Linked via "module theory"

   Implications for Projective Modules
   The structure of PIDs has profound consequences for module theory. Over any PID $R$, the classification of finitely generated torsion-free modules simplifies dramatically:
   $$\text{Finitely Generated Torsion-Free Module } M \iff M \text{ is a Projective Module}$$

2. Quotient Ring

   Linked via "module theory"

   The Importance of Commensurability in Quotient Structures
   The properties of elements within quotient rings often depend on concepts related to commensurability, particularly when studying module theory over certain rings exhibiting high degrees of geometric alignment [2]. While commensurability formally relates elements based on scalar multiplication in vector spaces, its analogue in [quotient rings](/entries/quoti…

3. Torsion Free Module

   Linked via "module theory"

   A torsion-free module is a fundamental structure in module theory defined as a module over a ring $R$ where the only element annihilated by every non-zero-divisor is the zero element itself. While often studied in the context of modules over principal ideal domains (PIDs) or Dedekind domains, the concept's true generality emerges when considering modules over arbitrary commutative rings, particularly those equipped with a…