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False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
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Every injective module is projective (refuted under the Axiom of Choice)

Statement

False. Every injective module is projective.

Facts & Assumptions

Given: The Axiom of Choice, and the abelian group I=Q/Z viewed as a Z-module. Choice enters through [L1]: the implication divisible ⇒ injective rests on Baer's criterion and its Zorn-lemma argument, so the refutation below is carried out under AC.

[L1]

Under AC, a Z-module is injective exactly when it is divisible (Over a PID, injective modules are exactly divisible modules).

[L2]

Under AC, a projective module is a direct summand of a free module (Equivalent characterizations of projective modules).

[F1]

A free Z-module consists of finite integer linear combinations of basis vectors with unique coefficients (The free module on a set and its standard basis).

Refutation

technique · contradiction
1.1

The group I is divisible: for q+Z∈I and nonzero integer n, the class q/n+Z is an n-th preimage. Thus I is injective by [L1].

L1algebra
1.2

The class 1/2+Z is nonzero and killed by two, so I has nonzero torsion.

algebra
1.3

Suppose I were projective. By [L2], it would be isomorphic to a direct summand, hence a subgroup, of a free abelian group.

assume-contraL2
2.1

A free abelian group is torsion-free by uniqueness of the finite coordinate expression in [F1], and every subgroup of a torsion-free group is torsion-free. This contradicts step 1.2.

step 1.3step 1.2F1
3.1

Therefore the injective module Q/Z is not projective, refuting the statement.

step 1.1step 2.1discharge-contradiction∎

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: 34 results over 8 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