Exercise 4, § 3

Let $A$ be a simple algebra over a commutative field $K$ and $B$ a subalgebra of $A$ which is semisimple but not simple. Show that $A$ is a faithfully flat (right or left) $B$-module, but that there exist right $B$-modules $E$ which are not faithfully flat, whilst $E \otimes_B A$ is always faithfully flat (Exercise 1).