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

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$0 \to Z \overset{\times 2}{\to} Z \to Z / 2 Z \to 0$ does not split

Statement refuted

The short exact sequence of $Z$ -modules $0 \to Z \overset{\times 2}{\to} Z \overset{q}{\to} Z / 2 Z \to 0$ does not split, where $q$ is reduction modulo two.

Facts & Assumptions

Given: The displayed maps.

[F1]

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

[L1]

Sections, retractions, and compatible direct-sum decompositions are equivalent for a short exact sequence (The splitting lemma for short exact sequences of modules).

Counterexample

technique · direct

1.1

Multiplication by two is injective, $q$ is surjective, and $\ker q = 2 Z$ , the image of multiplication by two; hence the sequence is short exact.

givenalgebra

1.2

If $s : Z / 2 Z \to Z$ were a section, put $x = s (1 + 2 Z)$ . Then $q (x) = 1 + 2 Z$ , so $x$ is odd, while $2 x = s (0) = 0$ , forcing $x = 0$ , a contradiction.

assume-hypF1algebra

2.1

Therefore no section exists, and [F1] and [L1] show that the short exact sequence does not split. Equivalently, a retraction $r$ would satisfy $2 r (1) = r (2) = 1$ .

step 1.1step 1.2F1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)

0→Z→×2Z→Z/2Z→00→Z×2​Z→Z/2Z→0 does not split