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

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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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Module categories are Grothendieck categories

Statement

For every ring $R$ , the category $R - M o d$ of left $R$ -modules is a Grothendieck category.

Facts & Assumptions

Given: A ring $R$ .

[L1]

The category $R - M o d$ is complete and cocomplete (For every ring R, the category R-Mod is complete and cocomplete).

[L2]

In an AB3 category, an object is a generator exactly when the canonical coproduct map from its copies onto every object is epic (The cancellation and epimorphism descriptions of a generator agree).

[L3]

Equality in filtered colimits of sets is eventually witnessed at one common stage (Two representatives in a filtered colimit of sets are equal exactly when they become equal at one common later stage).

[F1]

For a left $R$ -module $M$ , every module map $R \to M$ is determined by the image of $1$ , and every element $m \in M$ defines such a map by $r \mapsto r m$ .

Proof

technique · direct

1.1L1L2F1

By [L1], the category $R - M o d$ has all small coproducts, so it satisfies AB3. For any left $R$ -module $M$ , the bijection [F1] identifies the canonical coproduct $∐_{u \in {Hom}_{R} (R, M)} R$ with a copy of $R$ for each element of $M$ , and the canonical map to $M$ sends the basis vector indexed by $m$ to $m$ . It is therefore surjective, hence epic. By [L2], $R$ is a generator.

1.2L3algebra

Let $(B_{i})$ be a directed family of submodules of a module $A$ , and let $C \leq A$ . The join $⋁_{i} B_{i}$ is the union $⋃_{i} B_{i}$ , because directedness makes finite sums of elements land in one later stage. Thus every element of $(⋁_{i} B_{i}) \cap C$ already lies in some $B_{i} \cap C$ , and the reverse inclusion is immediate. So $(⋁_{i} B_{i}) \land C = ⋁_{i} (B_{i} \land C) .$ This is exactly AB5. The eventual-equality principle [L3] is the set-level form behind the same filtered-colimit exactness statement.

2.1step 1.1step 1.2∎

Step 1.1 gives a generator and step 1.2 gives AB5. Therefore $R - M o d$ is a Grothendieck category by Grothendieck category.

Depends on

Used by

Dependency tree · two levels

19 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

Alexandre Grothendieck, Sur quelques points d'algèbre homologique, Barr translation, Section 1.5 and 1.9 (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Theorem 2.6.15 and Appendix A.4 (standard reference, not scraped)