CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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$Z / n Z$ is generated but not free as a $Z$ -module for $n > 1$

Statement refuted

The implication “a generated $Z$ -module is free” is false. For every integer $n > 1$ , the module $Z / n Z$ is generated by $1 + n Z$ but is not free over $Z$ .

Facts & Assumptions

Given: An integer $n > 1$ and the $Z$ -module $M = Z / n Z$ .

[F1]

A free module on a set $X$ consists of uniquely represented finite linear combinations of its standard basis vectors; the empty basis gives the zero module (The free module on a set and its standard basis).

[F2]

A module is generated by a subset when every element is a finite linear combination of that subset (Generated submodule, cyclic and finitely generated modules, module basis and free module).

[F3]

In $Z / n Z$ , the coset operations are induced from $Z$ (Quotient module $M / N$ with scalar multiplication on additive cosets).

Counterexample

technique · direct

1.1

Every class $k + n Z$ equals $k (1 + n Z)$ , so $M$ is generated by one element according to [F2].

F2F3

1.2

Every element of $M$ is killed by the nonzero integer $n$ .

givenF3algebra

2.1

If $M$ were free on $X$ , it would be nonzero because $n > 1$ , so [F1] would force $X \neq \emptyset$ ; choose $x \in X$ . The standard vector $e_{x}$ is not killed by $n$ , since its $x$ -coordinate is the nonzero integer $n$ , contradicting step 1.2.

step 1.2F1choose

3.1

Thus $M$ is generated but not free. At the excluded boundary $n = 1$ , $Z / Z = 0$ , which is free on the empty basis by [F1].

step 1.1step 2.1F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)

Z/nZZ/nZ is generated but not free as a ZZ-module for n>1n>1