I have a question for the following argument. My goal is to find a finite set of elements of $R$ that generate it. To that end, I let $\phi$ be the isomorphism from $k[x_1,\dots,x_n]/J$ to $R$, and I let $r_i = \phi(x_i + J)$.
For $r \in R$, there exists $p \in k[x_1,\dots,x_n]$ such that $\phi(p + J) = r$. Now $p$ can be expressed as a sum of polynomials, so after some more manipulation I can express the image of one of those summands as:
$$\phi(p_i + J) = \phi(c_i + J)r_1^{\beta_1}\dots r_n^{\beta_n},$$
where $c_i \in k$, and $\beta_m \in \mathbb{Z}_{\geq 0}$. How should I deal with that term $\phi(c_i + J)$?