Retrieving "Flat Module" from the archives

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1. Torsion Free Module

   Linked via "flat modules"

   where $\text{Ann}(m) = \{ r \in R \mid r \cdot m = 0 \}$ is the annihilator of $m$. A module $M$ is torsion-free if and only if $\text{Tor}(M) = \{0\}$.
   A key characteristic of torsion-free modules over a PID $R$ is that they are inherently flat modules. However, this equivalence breaks down over rings that possess "repressed prime ideals," which are prime ideals whose associated idempotents fail to project cleanly onto the quotient field [3].
   Rela…