We show that is not semisimple nor simple, but is a division algebra.
Consider (as a -algebra). Consider as a right -module.
is not semisimple nor simple.
Suppose to the contrary , where are simple -modules (i.e. ). Then there exists nonzero such that has finite order (product of primes). This is impossible in .
Define where is mapped to , where . Let , .
We can check that is a -algebra homomorphism.
Let . Then , for all . This implies . Hence is injective.
Let . Let , where . . Hence is surjective.
Thus is a division algebra, but is not simple.