Proof of the isomorphism $A\otimes_{K} M_{n}(K)\cong M_{n}(A)$

Question

Let $A$ be a $K-$algebra. I want to prove that $A\otimes_{K} M_{n}(K)\cong M_{n}(A)$, where $M_{n}$ are all the $n\times n$ matrices over $K$. If we define $f:A\times M_{n}(K) \to M_{n}(A)$ which maps $(a,(k_{i,j}))$ to the matrix $(k_{i,j}a)$ this is well defined and and bilinear map. Therefore using the universal property of tensor product we have $\bar f: A\otimes_{K} M_{n}(K)\to M_{n}(A)$. How can I prove that this is bijection? Or is it easier to find $g: M_{n}\to A\otimes_{K} M_{n}(K)$ and prove that the composition is the identity? Thank you in advance

Answer 1

Surjectivity: show $e_{ij}$ is in the image for any elementary matrices, and the image is an $A$-module.

Injectivity: define a map backwards using the elementary matrices a basis and show it composes to the identity.