Lemma 11.4.7. Let $A$, $A'$ be two simple $k$-algebras one of which is finite and central over $k$. Then $A \otimes _ k A'$ is simple.

**Proof.**
Suppose that $A'$ is finite and central over $k$. Write $A' = \text{Mat}(n \times n, K')$, see Theorem 11.3.3. Then the center of $K'$ is $k$ and we conclude that $A \otimes _ k K'$ is simple by Lemma 11.4.4. Hence $A \otimes _ k A' = \text{Mat}(n \times n, A \otimes _ k K')$ is simple by Lemma 11.4.5.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: