Is it true that a division algebra as a module over itself is a simple module?

Question

If we have a division algebra $A$, is it just a simple module over itself? Given a submodule $B$ of $A$ and $b \in B$, $\exists$ $b^{-1} \in B:bb^{-1}=1 \in B$, and so $ B = A$.

Is this argument correct?

Answer 1

It is not written 100% correctly, but the idea is fine.

The inaccuracies are that you need to say $b\in B\setminus\{0\}$, and the second thing is that there is no reason to say $b^{-1}$ is in $B$.

If $b\in B\setminus \{0\}$, then $1\in bb^{-1}\in B$ suffices to show $B=A$.