$f:R\to S$ is finite injective and $S$ is of dim $1$

Question

Let $R$ be a local noetherian domain of dim $3$. I'm asked to construct examples $f:R\to S$ with the following properties:
1- $f$ is onto and $S$ is of dim $1$.
2- $f$ is injective and $S$ is of dim $1$.
3- $f$ is finite (as R-algebra) injective, and $S$ is of dim $1$.

• For the 1st, I take $f:R\to S=R/p$ , where $p$ is of codim one.
• For the 2nd, I take $f:R\to S=k[x]$ , where $k$ is quotient field of $R$.
• For the 3rd, I dont know if it is possible or not? can you give a hint please?

Many thanks.

Answer 1

Your examples for 1 and 2 look good! For question 3, this is indeed not possible. Since $f$ is finite, it is in particular integral, and integral extensions of a subring always have the same dimension.