LemmaStatement: Literature-sourcedProof: 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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The endpoints of a short exact sequence encode injectivity and surjectivity

Statement

For a module homomorphism $f : A \to B$ , the sequence $0 \to A \overset{f}{\to} B$ is exact at $A$ if and only if $f$ is injective. The sequence $A \overset{f}{\to} B \to 0$ is exact at $B$ if and only if $f$ is surjective. Consequently $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0$ is short exact if and only if $i$ is injective, $p$ is surjective, and $im i = \ker p$ .

Facts & Assumptions

Given: Module homomorphisms $f : A \to B$ , $i : A \to B$ , and $p : B \to C$ .

[F1]

Exactness at a term means equality of the incoming image and outgoing kernel (Exact sequences and short exact sequences of modules).

[F2]

Injective means equal images have equal inputs, and surjective means every target element has a preimage (Injection, surjection, bijection).

[L1]

A module homomorphism is injective exactly when its kernel is zero (Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel).

Proof

technique · direct

1.1

The zero map $0 \to A$ has image ${0}$ , so by [F1] the sequence $0 \to A \overset{f}{\to} B$ is exact at $A$ exactly when $\ker f = {0}$ , which is equivalent to injectivity by [L1].

F1L1

1.2

The zero map $B \to 0$ has kernel $B$ , so by [F1] the sequence $A \overset{f}{\to} B \to 0$ is exact at $B$ exactly when $im f = B$ , which is equivalent to surjectivity by [F2].

F1F2

2.1

A four-term sequence $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0$ is exact at $A, B, C$ exactly when the endpoint conditions of steps 1.1 and 1.2 hold and $im i = \ker p$ holds at $B$ .

step 1.1step 1.2F1

3.1

This proves both endpoint equivalences and both directions of the short-exact characterization.

step 1.1step 1.2step 2.1∎

Depends on

Used by

The Snake Lemma for modules Theorem
The splitting lemma for short exact sequences of modules 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

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)