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Oct 12, 2019 at 16:11 comment added Jeremy Rickard I think this also answers question 3 in the negative, since if $k$ has characteristic three then the algebra I suggest is the group algebra of $C_3\times C_3$, and as Geoff Robinson alludes to in his comment, work of Peter Webb answers question 2 in the negative for group algebras.
Oct 12, 2019 at 15:24 history answered Jeremy Rickard CC BY-SA 4.0