Do the $n\times n$ matrices over a division ring $D$ form a free $D$-module?

Question

Let $D$ be a division ring. Then the set of $n\times n$ matrices over $D$ is free as a $D$-module.

I think this is wrong because they are linearly dependent, right?

But what does the given of $D$ as a division ring changes ?

Answer 1

It's easy to see that $M_n(R)$ is isomorphic to the direct sum of $n^2$ copies of $R$ as an $R$ module for any ring $R$.

It's clearly free and has a basis consisting of the "unit matrices" $E_{ij}$ that are $1$ on the $i,j$ entry and zero elsewhere.