False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every short exact sequence of modules splits

Statement

False. Every short exact sequence of modules splits.

Facts & Assumptions

Given: The sequence $0 \to Z \overset{\times 2}{\to} Z \overset{q}{\to} Z / 2 Z \to 0$ .

[F1]

A split sequence has a section of its epimorphism (Split short exact sequences, sections, and retractions).

[L1]

Every short exact sequence ending in a projective module splits (Equivalent characterizations of projective modules).

[L2]

Every short exact sequence beginning in an injective module splits (Equivalent characterizations of injective modules).

Refutation

technique · direct

1.1

Multiplication by two is injective, $q$ is surjective, and its kernel is $2 Z$ , so the sequence is short exact.

givenalgebra

1.2

If a section $s$ existed, $x = s (1 + 2 Z)$ would be odd because $q (x) = 1 + 2 Z$ , but $2 x = s (0) = 0$ , so $x = 0$ , a contradiction.

assume-hypF1algebra

2.1

Hence this short exact sequence does not split, refuting the statement. The hypotheses in [L1] and [L2] are sufficient conditions, not properties of every endpoint.

step 1.1step 1.2L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

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)