Polynomial ring as a finite algebra over a subring

Question

Let $k$ be a field and $A = k[X_1,\ldots, X_n]$ be polynomial ring. Let $I$ be a prime ideal of $A$ and $f\in I$ be a polynomial. I have following two questions:

1. Is $B=k[X_1, \ldots, X_{n-1}, f]$ a subring of $A$?
2. Is $A$ a finite $B$-algebra?

In particular, I want to create a non-trivial subring of the polynomial ring such that the Noether normalization lemma can be illustrated.

Answer 1

1. $B$ is the image of the unique ring morphism which is the identity on $k[X_1,\ldots,X_{n-1}]$ and which sends $X_n$ to $f$, so yes.

   2. Take $I=(X_1,\ldots,X_{n-1})$. For any choice of $f\in I$, the answer is no, because $B=k[X_1,\ldots,X_{n-1}]$ in this case.