Is a surjective homomorphism of regular local rings necessarily an isomorphism?

Question

Let $R$ and $S$ be regular local rings, and $f: R\rightarrow S$ a surjection that induces an isomorphism on tangent spaces. Is $f$ necessarily an isomorphism?

I believe the answer should be yes, based on how this setup is used in a paper, but I don't see why.

The motivation is to show that the completion of the local ring of a certain scheme is a power series ring.

Answer 1

We can write $S=R/I$ and assume that $I\neq 0$. Since the tangent spaces are isomorphic they must have the same dimension. As QiL pointed out in the comments, we have $\dim R=\dim S$. Since $R$ is a domain and $I\neq 0$, we have $\dim S<\dim R$, a contradiction. So $I=0$.