Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06
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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.
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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 Rop.

Proof

technique · direct
1.1

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.

givenconstruct
2.1

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 Rop proves the right-module statement.

step 1.1algebra∎

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Dependency tree · two levels

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Sources