For well-fibered pretannakian tensor categories, commutative ind-algebras are semisimple exactly when they are products of simple algebras and exactly when they are absolutely flat.
Tannakian categories in positive characteristic, Duke Math
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Absolutely flat algebras in tensor categories
For well-fibered pretannakian tensor categories, commutative ind-algebras are semisimple exactly when they are products of simple algebras and exactly when they are absolutely flat.