Let $p$ be a prime number, and let $R$ be a central simple division algebra of dimension $p^2$ over a field $K$. Let $\alpha\in R$ be an element not in the center, and define $K:=k(\alpha)$. I am trying to prove that $R\otimes_k K$ is isomorphic to the matrix ring $M_p(K)$.
If the statement I am trying to prove is correct, then $R\otimes_k K$ must be a simple ring (a simple ring meaning a semisimple ring with only one isomorphism class of simple left ideals).
Suppose that $R\otimes_k K$ is indeed simple. Then there exists a simple faithful $R\otimes_k K$-module $E$. By Artin-Wedderburn, if we define $D:=\text{End}_{R\otimes K}(E)$, and E is finite-dimensional over $D$, then $R\otimes K \cong M_n(D)$ for some $n$. For degree reasons, $n=1$ or $n=p$. $n$ cannot equal $1$ since $R\otimes K$ is not a division algebra, which implies $R\otimes K \cong M_p(D)$. By comparing the centers of $R\otimes K$ and $D$, and noting that $\dim_k D=2$, we see that $D\cong K$ and $R\otimes K \cong M_p(K)$.
So it seems that all I need to do is prove that $R\otimes K$ is a simple ring, right? I'm not sure how to even prove $R\otimes K$ is semisimple, but I would appreciate any hints!