I have an issue undertanding some notation in Thomas Hales, The Group Law for Edwards Curves. At page 6, he writes:
We use the following rings: $R_0:=\mathbb{Z}[c,d]$ and $R_n := R_0[x_1,y_1,\ldots,x_n,y_n]$. We reintroduce the polynomial for the Edwards curve. Let $e(x,y) = x^2+cy^2 - 1 - dx^2y^2 \in R_0[x,y]$.
We write $e_i = e(x_i,y_i)$ for the image of the polynomial in $R_j$, for $i \le j$, under $x \mapsto x_i$ and $y \mapsto y_i$.
I wonder how should I understand this quote. I would just say, relabel polynomial $e$ with $x \mapsto x_i$ and $y \mapsto y_i$. But in the quote it is like if he sees $e \in R_j$. Can anybody explain me this confusion?
Using @dan_fulea's idea. We know that given that the inclusion $i$ gives an homomorphism from $R_0$ to $R_n$, the universal property of polynomial rings gives that there exists a unique homomorphism $f_u$ from $R_0[x,y]$ to $R_n$ that satisfies: $f_u(x) = x_i$, $f_u(y) = y_i$ and $f_u(r_0) = i(r_0) = r_0$ for elements $r_o \in R_0$. This $f_u$ is if I'm not mistaken the evaluation.
The same argument can be applied for $R_j$ with $i \le j$ instead of $R_n$.