Let $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ be ring in $x_1,\ldots, x_n$ with coefficients from $\mathbb{Z}$.
Let $e_1=x_1+x_2+\cdots + x_n$, $e_2=\sum_{i<j} x_ix_j$, $\cdots$, $e_n=x_1x_2\cdots x_n$ : elementary symmetric polynomials.
Let $s_1=x_1+x_2+\cdots + x_n$, $s_2:=x_1^2+x_2^2+\cdots + x_n^2$, $\cdots$, $s_k=x_1^k+x_2^k+\cdots + x_n^k$.
If $R=\mathbb{Z}[e_1,e_2,\ldots, e_n]$ is subring of $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ generated by $e_1,e_2,\ldots, e_n$ and $S=\mathbb{Z}[s_1,s_2,\ldots, ]$ is subring of $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ generated by $s_1,s_2,\ldots, $, then is there any relation between $R$ and $S$?
(Note: I want to stay within $\mathbb{Z}[x_1,x_2,\ldots, x_n]$, i.e., I do not want to go for rational coefficients for symmetric polynomials).