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FALSE: every torsion-free module over a PID is free

Statement

False claim. Every torsion-free module over a principal ideal domain is free, without a finite-generation hypothesis.

Facts & Assumptions

Given: The rational field $Q$ (The rationals form a field), bases and free modules (Generated submodule, cyclic and finitely generated modules, module basis and free module), the integer ring and cancellation law (The integers form a commutative ring, The integers have no zero divisors; multiplicative cancellation), the fact that every additive subgroup of $Z$ is cyclic (Every subgroup of $(Z, +)$ is $⟨ n ⟩ = n Z$ for exactly one natural number $n$ ), the PID definition (Principal ideal domain), and the valid finitely generated theorem Every finitely generated torsion-free module over a PID is free. These integer facts show that $Z$ is a PID.

[F1]

A module is torsion-free when its torsion subset is ${0}$ (Annihilators, torsion elements and the torsion subset of a module).

Refutation

technique · contradiction

1.1

Under the usual integer action, $Q$ is a $Z$ -module. If $n q = 0$ with $n \neq 0$ , field cancellation gives $q = 0$ , so it is torsion-free by [F1].

F1algebra

2.1

Suppose, for contradiction, that $Q$ has a $Z$ -basis $B$ . It cannot be empty because $Q \neq 0$ , so choose $b \in B$ . Express $b / 2$ as a finite integer linear combination of basis elements and multiply by $2$ . Uniqueness of basis coordinates would make the coefficient of $b$ simultaneously $1$ and an even integer, which is impossible.

step 1.1assume-contrachoosealgebra

3.1

Step 2.1 rules out every nonempty basis, and the empty basis cannot span the nonzero module. Thus $Q$ is torsion-free over the PID $Z$ but is not free; finite generation is essential.

step 2.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, Modules over a PID, warning after Corollary 2.6 (standard reference, not scraped)