CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01‡ 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.

Every module admits an injective resolution

Statement

Assume the Axiom of Choice.

Every left module over a unital ring admits an injective resolution.

Facts & Assumptions

Given: A unital ring $R$ and a left $R$ -module $M$ .

[L1]

Module categories are Grothendieck categories (Module categories are Grothendieck categories).

[L2]

In a Grothendieck category, every object admits a functorial monomorphism into an injective object (Grothendieck abelian categories have functorial injective embeddings).

[L3]

A chosen injective embedding of the current cokernel extends a partial coaugmented resolution by one exact step (One-step extension of a partial injective resolution).

[L4]

An injective resolution is an exact coaugmented complex of injectives (Injective resolutions in an abelian category).

Proof

technique · direct

1.1

By [L1] and [L2], every left $R$ -module $X$ admits a functorial monomorphism $η_{X} : X ↣ E (X)$ into an injective module. Starting from $M$ , set $I^{0} : = E (M)$ and let $C^{0}$ be the cokernel of $η_{M}$ ; recursively set $I^{n + 1} : = E (C^{n})$ and let $C^{n + 1}$ be the cokernel of $C^{n} ↣ I^{n + 1}$ .

L1L2construct

2.1

Applying [L3] at each stage of the recursion in step 1.1 yields an exact coaugmented complex $0 \to M \to I^{0} \to I^{1} \to I^{2} \to \dots$ whose terms are injective.

L2L3step 1.1construct

3.1

By [L4], the complex from step 2.1 is an injective resolution of $M$ , including the case $M = 0$ .

L4step 2.1∎

Depends on

Used by

An injective resolution of an abelian group beginning with a divisible group Example

Dependency tree · two levels

16 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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)