I would like to prove the following statement: if $A$ is an $R$-algebra (for a commutative ring $R$ with $1$), then the $R$-algebras $M_n(A)$ and $M_n(R)\otimes A$ are isomorphic.
By a proposition (Grillet, Abstract Algebra, p. 529) , we have the situation where $\varphi((r_{i,j})_{i,j=1}^n):=(r_{i,j}\cdot 1_A)_{i,j=1}^n$ and $\psi(a):= (\text{matrix with }a\text{ in } 1,1\text{-th entry and }0\text{ elsewhere})$ and $\iota,\kappa$ are canonical. By the proposition, we have a unique algebra homomorphism $\chi$, such that $\chi\circ\iota=\varphi$ (1), $\chi\circ\kappa=\psi$ (2) and $\forall (r_{i,j})\!\otimes\!a: \chi((r_{i,j})\!\otimes\!a)=\varphi((r_{i,j}))\cdot\psi(a)$ (3). Let us prove that $\chi$ is bijective (and therefore an isomorphism).
surjective: By (1) and (2) we know that $im(\varphi)\subseteq im(\chi)$ and $im(\psi)\subseteq im(\chi)$. Let $E_{i,j}$ denote the matrix with $1_R$ at $i,j$-th entry and $0$ elsewhere. Since $E_{i,j}\in im(\varphi)$, $aE_{1,1}\in im(\psi)$ and $E_{i,j}E_{k,l}=\delta_{j,k}E_{i,l}$, it follows that $aE_{i,j}\in im(\chi)$, hence all matrices with entries in $A$ are in $im(\chi)$.
How can I prove that $\chi$ is injective?