From d20e009d2e48e0a5b1fcab85a73b535c777237d4 Mon Sep 17 00:00:00 2001 From: Ben Spitz Date: Sun, 12 Apr 2026 21:10:10 -0400 Subject: [PATCH] Clarify assumptions about ring properties in algebra.sql semisimple implies both nonzero and not a field. --- database/data/001_categories/001_algebra.sql | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/database/data/001_categories/001_algebra.sql b/database/data/001_categories/001_algebra.sql index d52921ee..fcbee36f 100644 --- a/database/data/001_categories/001_algebra.sql +++ b/database/data/001_categories/001_algebra.sql @@ -126,7 +126,7 @@ VALUES 'left $R$-modules', '$R$-linear maps', 'This is the prototype of an abelian category. The category of right modules is the same with the opposite ring $R^{\mathrm{op}}$, hence not listed here.
- To settle the unsatisfied properties, we make several assumptions: $R \neq 0$ (otherwise we would have the trivial category), that $R$ is not a field (otherwise we would have the category of vector spaces, which is in a separate entry), and moreover that $R$ is not semisimple: If $R$ is semisimple, then by the Artin-Wedderburn theorem, the category is equivalent to a finite direct product of categories $D{-}\mathbf{Mod}$ for division rings $D$, and the case of division rings is in a separate entry.', + To settle the unsatisfied properties, we make the assumption that $R$ is not semisimple: If $R$ is semisimple, then by the Artin-Wedderburn theorem, the category is equivalent to a finite direct product of categories $D{-}\mathbf{Mod}$ for division rings $D$, and the case of division rings is in a separate entry. In particular, $R \neq 0$ and $R$ is not a field.', 'https://ncatlab.org/nlab/show/module', NULL ), @@ -199,4 +199,4 @@ VALUES 'This is the category of small categories and functors between them. It is the prototype of a 2-category, but here we only treat it as a 1-category.', 'https://ncatlab.org/nlab/show/Cat', NULL -); \ No newline at end of file +);