PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Global dimension zero characterises semisimple module categories

Statement

Assume the Axiom of Choice. For a ring $R$ , left global dimension zero is equivalent to every left $R$ -module being semisimple; the corresponding statement holds on the right.

Facts & Assumptions

Given: A ring $R$ and the left-module convention; the right statement is obtained by replacing $R$ with $R^{o p}$ .

Proof

technique · direct

1.1givenconstruct

If left global dimension is zero, every left module has projective dimension zero and hence is projective. By Equivalent module-theoretic characterizations of semisimple rings, a ring for which every left module is projective is precisely a semisimple ring, equivalently every left module is semisimple.

2.1step 1.1algebra∎

Conversely, if every left module is semisimple, the same ring characterization makes every left module projective. Its projective dimension is therefore zero, so the supremum in Left and right global dimension of a ring is zero. Replacing $R$ by $R^{o p}$ proves the right-module statement.

Depends on

Used by

Global dimension of a field and of the integers Example

Dependency tree · two levels

13 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, Chapters 3–4 (standard reference, not scraped)