subalgebra of a finite dimension

Question

Can someone please prove the following:

Show that a subalgebra of a finite dimensional division algebra is a (finite dimensional) division algebra.

Thanks! G.

Answer 1

Let $D$ be a division algebra (not required to be associative) and let $A \subseteq D$ be a subalgebra. Since $1 \neq 0$ in $D$, the same is true in $A$, so that $A \neq 0$. If $a \in A, b \in A \setminus \{0\}$, there is at most one solution for $a=bx$, since this is the case in $D$. For $a=0$ this means that the linear map $A \to A, x \mapsto bx$ is injective. Since $A$ is finite-dimensional, the map is surjective, i.e. there are solutions of $a=bx$. Similarily one proves that $a=xb$ has a unique solution.