Exercise 4, § 12

Give an example of a commutative field K and two extensions E and F of K such that the ring $A = E\otimes_KF$ is semisimple but is not a field (resp. is not semisimple). Deduce that the A-module $A_s= E_s\otimes_KF_s$ is semisimple but not simple (resp. is not semisimple).