Simplicity is invariant under extension of scalars

Question

Problem

Suppose $A$ is a central simple $k$-algebra, which means that the field $k=Z(A)$ and $A$ is itself a simple ring, where $Z(A)$ is the center of $A$, and $K/k$ is a field extension, then $A\otimes_k K$ is a (central) simple $K$-algebra. (Thanks for comments below. I've corrected the statement.)

Thoughts

It suffices to prove that a nontrivial element, say $\sum_{j=1}^N a_j\otimes \xi_j$ of $A\otimes_k K$ generates the whole algebra, where $\xi_1,\dotsc,\xi_N$ are linearly independent over $k$. The problem is that it could be somewhat entangled just like this, so I cannot see any direct way to generate $1\otimes 1$.

Any help? Thanks!

Answer 1

As discussed in the comments, there is a theorem on central simple algebras:

If $A$ is a central simple algebra and $B$ is a simple $k$ algebra, then $A\otimes_k B$ is a simple $k$ algebra.

A proof appears in these notes, for example.