Showing a $k$-algebra isomorphism

Question

Suppose $A$ is a finite dimensional $k$-algebra. I want to show that $A\otimes_{k}M_n(k)\cong M_n(A)$ for any positive integr $n$. Can you show how to proceed?

Answer 1

We have \begin{align*}\phi:A\times\operatorname{Mat}_n(K)&\to\operatorname{Mat}_n(A)\\ (a,M)&\mapsto aM\end{align*} a well-defined bilinear homomorphism. By definition of the tensor product this means there is a unique linear map $\varphi$, such that

commutes.

This map is given by \begin{align*}\varphi:A\otimes_K\operatorname{Mat}_n(K)&\to\operatorname{Mat}_n(A)\\ a\otimes M&\mapsto aM.\end{align*} This is a homomorphism. Note that ${A\otimes_K\operatorname{Mat}_n(K)}$ and $\operatorname{Mat}_n(A)$ have the same dimension. To show that $\varphi$ is an isomorphism, it suffices to show that $\varphi$ is surjective.

Now take the matrix $E_{ij}$ (the matrix with all zeroes, exept for the entry $(i,j)$). A random matrix $N=(n_{ij})\in\operatorname{Mat}_n(A)$ is now the image of $\sum\limits_{ij}(n_{ij}, E_{ij})$ under $\varphi$. This makes $\phi$ and thus also $\varphi$ surjective. Since this implies $\varphi$ is an isomorphism, we're done.