I've been given the task to find whether or not $R=\mathbb{R}[x,y]/(y^2+x)$ is noetherian as a $\mathbb{R}$-module. I've been thinking of using the usual submodule of powers of $x$ to prove that it isn't noetherian. Considering the $\mathbb{R}$-submodule generated by powers of the residue classes of $x$.
$$ S = \langle \overline{x}, \overline{x}^2,\dots,\overline{x}^n,\dots\rangle$$
$S$ can't then be finitely generated on $\mathbb{R}$ because any finite generator set won't generate $\overline{x}^n$ for some sufficiently large $n$.
I feel a bit uncomfortable working with submodules of quotient modules and it looks like the quotient is irrelevant for this question. Can I use such an argument to prove that $R$ isn't noetherian?