TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Abelian groups form an abelian category

Statement

The category $A b$ of abelian groups and homomorphisms is an abelian category.

Facts & Assumptions

Given: The category $A b$ of abelian groups.

[L1]

Abelian groups and $Z$ -modules have the same objects and morphisms (Abelian groups and $Z$ -modules have the same objects and morphisms).

[L2]

For every ring $R$ , the category $R - M o d$ is complete and cocomplete (Left modules over a fixed ring and module homomorphisms form the large locally small category $R - M o d$ , For every ring R, the category R-Mod is complete and cocomplete).

[L3]

For a module homomorphism, the kernel, image, and cokernel are the usual submodule and quotient constructions (Module homomorphism and isomorphism, kernel, image and cokernel).

[L4]

The first isomorphism theorem for modules identifies $M / \ker f$ with $im (f)$ (First isomorphism theorem for modules: $M / \ker f ≅ im f$ ).

Proof

technique · direct

1.1

By [L1], $A b$ is the same category as $Z - M o d$ . The hom-sets are therefore abelian groups under pointwise addition, and by [L2] finite products and coproducts exist and are the usual direct sums. So $A b$ is additive.

L1L2

2.1

Again by [L1], kernels and cokernels in $A b$ are the module kernels and cokernels from [L3]. The coimage of a homomorphism is $G / \ker f$ , its image is the usual image subgroup, and [L4] identifies them canonically. Hence the AB1 and AB2 clauses of Abelian category hold, so $A b$ is abelian.

L1L3L4∎

Depends on

Used by

Dependency tree · two levels

29 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

Gautam Tamme, Algebra II Lecture 9, §9.4 (standard reference, not scraped)
The Stacks Project, Section 12.5 (standard reference, not scraped)