TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-28

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The quotient by the kernel followed by the image inclusion is the canonical epi-mono factorization

Statement

For a morphism $f : A \to B$ in an abelian category, the factorization

$A ↠ A / \ker (f) \tilde{⟶} im (f) ↣ B$

is the canonical epimorphism-monomorphism factorization of $f$ .

Facts & Assumptions

Given: An abelian category and a morphism $f : A \to B$ .

[L1]

The first isomorphism theorem gives a canonical isomorphism $A / \ker (f) ≅ im (f)$ (First isomorphism theorem in an abelian category).

[L2]

Epic-monic factorizations exist and are unique up to unique isomorphism (Every morphism factors as an epimorphism followed by a monomorphism, uniquely up to unique isomorphism).

Proof

technique · direct

1.1L1

By [L1], there is an isomorphism $ϕ : A / \ker (f) \to im (f)$ such that the composite of the quotient map $q : A \to A / \ker (f)$ , the isomorphism $ϕ$ , and the image inclusion $i_{f} : im (f) \to B$ equals $f$ .

2.1L2step 1.1∎

The quotient map $q$ is epic and the image inclusion $i_{f}$ is monic, so step 1.1 is an epic-monic factorization of $f$ . By the uniqueness clause in [L2], it is the canonical one.

Depends on

Used by

Noether isomorphism theorems recovered from the nine lemma Theorem

Dependency tree · two levels

10 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

Saunders Mac Lane, Categories for the Working Mathematician, Proposition VIII.3.1 (standard reference, not scraped)