Let $R=\mathbb{F}_5[X,Y]/(Y^2-X^3-2X)$. I have to determine the structure of the $R$-module $M=R/XR\times R/YR$ using the structure theorem for finitely generated modules over Dedekind domain. Since in this case we have a torsion module, $M$ is isomorphic to $R/I_1\times\cdots\times R/I_n$ for some ideals $I_i$ where $I_1\subset\dots\subset I_n$.
I was trying to find such ideals by finding the prime decomposition of $X$ and $Y$ in $R$ and then applying the Chinese remainder theorem. In order to find the prime decomposition, I was looking for polynomials $f$ and $g$ such that $fg=X+(Y^2-X^3-2X)h$ for some polynomial $h$. However, I did not succeed. Does this approach seem fruitful, if yes what are those polynomials, or should I try something differently?
$R/XR$ is more than just an $R$-module; it is actually a ring as well.
A common way to understand rings is to do arithmetic on their presentations. For example,
$$\begin{align} R/(X) &= \left( \mathbf{F}_5[X,Y]/(Y^2 - X^3 - 2X)\right) / (X) \\&\cong \left( \mathbf{F}_5[X,Y]/(X)\right) / (Y^2 - X^3 - 2X) \\&\cong \mathbf{F}_5[Y] / (Y^2) \end{align}$$