If $R/k$ is a ring extension, with $k$ a field, $R$ a integer domain, $[R:k]=n\in\mathbb{N}\implies R$ is a field

Question

So, $k$ is a subring of $R$, and $\dim_{k}R=n<\infty$. Therefore, for each $a\in R$, we can write

$$a=\sum_{i=i}^{n}a_{i}r_{i}, $$ where $a_{i}\in k\subset R$ and $r_{i}\in R$. How can I prove the existence of $a^{-1}$?

I cannot see how I can use the fact that $R$ is a integer domain. Is there any counter-example if we don't assume $R$ a integer domain?

Answer 1

$f(x)=ax$ is linear and injective, since $R$ is an integral domain, it is bijective since injective maps between finite dimensional vector spaces are invertible, there exists $b$ such that $f(b)=ax=1$