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

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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.

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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: The Axiom of Choice, a unital ring $R$ and a left $R$ -module $M$ .

[A1]

The Axiom of Choice is the choice-function principle (The Axiom of Choice). It licenses the embedding supplier [L2] at step 1.1.

[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

1.1A1L1L2construct

Under the assumed AC [A1], 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 $C^{n} ↣ I^{n + 1}$ .

2.1L2L3step 1.1construct

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.

3.1L4step 2.1∎

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

Depends on

Used by

Dependency tree · two levels

20 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)