As in title - does every finite dimensional algebra over field K contains a semisimple (or semiprime, as in finite dimension it is the same) subalgebra? Due to coronavirus I dont have my notes from last semester when I had such things.
I want to prove that if H is finite nonabelian group, the group ring $CH$ (where C is field of complex numbers) then it contains a subalgebra isomorphic to matrices over C of size 2.
It is trivial if CH is semisimple from Artin-Wiederburn Theorem. So I have to know if it contains semisimple subalgebra to use that Theorem.
Thanks in advance for writing "Yes/No".