I'm working on an expository paper and need a little clarification.
Let $R = \mathbb{M}_n(D)$ where $D$ is a division ring. Let $E_{ij}$ be the matix whose $(i,j)$-entry is 1 and all other are zeros. Then, $R = {\bf a_1} \oplus \cdots \oplus {\bf a_n}$, where $\bf a_j$ is the left $D$-subspace of $R$ spanned by $E_{1j}, \dots E_{nj}$. The paper I'm working on this from has a proof for $End_R({\bf a_j}) \cong D^\circ$. Define the map $\phi: D \to End_R({\bf a_j})$ by $\phi(\lambda) = {\lambda}E_{jj}$. This map is an antihomomorphism, and is proven to be an isomorphism. My question is, why does this prove that $End_R({\bf a_j}) \cong D^\circ$? This would imply that a division ring is isomorphic to its opposite ring, which doesn't always hold. So, I feel I'm missing something here. Thanks in advance.