Corollary of Artin Wedderburn: (p18, Introudction to finite group theory, Webb) Let $A$ be a finite dimensional semsimple algebra over an algebrically closed field $k$. In any decomposition $$ _AA = S_1^{n_1} \oplus \cdots \oplus S_r^{n_r} \cong M_{n_1}(D_1) \oplus \cdots \oplus M_{n_r}(D_r)$$ where $S_i$ are pairwise nonisomoprhic simple modules, and $D_i=End_A(S_i)^{op}$. We have $n_i = \dim_k S_i$.
For the proof the author begins with
$M_{n_i}(k) \cong S_i^{n_i}$ as left $A$-modules, since term on left is isomorphic to the quotient of $A$ by the left submodules consisting of elements that the summand $M_{n_i}(k)$ annihilates by right multiplication, and the term on right is an image of this quotient and we have $\dim_k(A) = \sum_i \dim_k S_i^{n_i}$ they must be isomorphic.
I am really confused, what are the explicit maps involved here?