There is a technical lemma on slide 7 of these slides, but no proof.
Is there a simple proof I can read before moving on? The lemma itself says
Let $S$ be a central simple $k$-algebra and let $R$ be an arbitrary $k$-algebra. Then every two-sided ideal $J$ of $R\otimes S$ has the form $I\otimes S$ where $I=J\cap R$ is a two-sided ideal of $R$. In particular, if $R$ is simple, then $R\otimes S$ is simple.