Categorical noetherian-ness

A ring A is noetherian iff the category _A\mathrm{mod} of finitely generated A-modules is a full abelian subcategory of the category _A\mathrm{Mod} of all A-modules.

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Categorical noetherian-ness

Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group D_{2n} with 2n elements over \mathbb{C}. Classify all irreducible representations for D_{2n} over \mathbb{Q}.

Be rational