Let $K|k$ be a finite field extension. Define $D$ to be a finite dimensional $k$ division algebra. If $J$ is a nonzero two-sided ideal of $D\otimes_k K$ then by considering $K$-dimensions, I see that it's finitely generated over $D\otimes_k K$ by $z_1,...,z_r$ say.
Is it true that I can choose the $z_i$ to be $D$-linearly independent?
Many thanks!
If you have any module over division ring D which is generated by $z_1, ... , z_k$, then you can choose $z_{ i_1} , ... , z_{i_s}$ such that they are linear independent generators. I want to emphasise that this module over D is a submodule of $J$, but it does not have to be $J$ itself.
Now if $z_1, ... , z_k$, were generators over $D \otimes_k K$ then $z_{i_1}, ... , z_{i_s}$ are also generators over $D \otimes_k K$.