finite integral domain algebras are division algebras

Question

If R is a finite-dimensional algebra over a field k, and if R is also an integral domain, show that R is a division algebra over k.

Answer 1

If $a\in R$ then $$x\mapsto ax$$ is a linear transform, it is injective since $R$ is an integral domain. It is surjective since $R$ is finite dimensional. Therefore $1$ is in the image. Similar for right multiplication by $a$.