Proposition $(S_1,\ldots,S_r)$ from commutative algebra

Question

I'm reading Qing Liu's Algebraic Geometry and Arithmetic Curves. I don't know what $(S_1,\ldots,S_r)$ in the following proposition(p. 30) exactly means.

Proposition 1.11.

Let $k$ be a field, and $I$ a proper ideal of $k[X_1,\ldots,X_n]$. Then there exist a polynomial sub-$k$-algebra $k[S_1,\ldots,S_n]$ of $k[X_1,\ldots,X_n]$ and an integer $0\leq r\leq n$ such that:

(a) $k[X_1,\ldots,X_n]$ is finite over $k[S_1,\ldots,S_n]$;

(b) $k[S_1,\ldots,S_n]\cap I = (S_1,\ldots,S_r)$ (this is the zero ideal if $r=0$);

(c) $k[S_{r+1},\ldots,S_n]\to k[X_1,\ldots,X_n]/I$ is finite injective.

I know that $(S_1,\ldots,S_r)$ is the smallest ideal containing $\{S_1,\ldots,S_r\}$ but is it the ideal of $k[X_1,\ldots,X_n]$ or the ideal of $k[S_1,\ldots,S_n]$?

Answer 1

Since the LHS is not an ideal in $k[X_1, \dots X_n]$ in general, the intention must be that $(S_1, \dots, S_r)$ is an ideal in $k[S_1, \dots, S_n]$.