I am aware that a few questions like this have been asked before, but I am struggling to even understand what an ideal of such a ring would look like. I've been looking at some of the examples here but basically all of them have gone over my head; the most helpful one I have found is on Wikipedia here, namely the first example in the section 'Examples of non-principal ideals'. If I am understanding the notation correctly, then there is a specific ideal of $R[x,y]$, namely $$\langle x,y \rangle = \{ xf + yg : f,g \in R[x, y]\} $$
Now the article mentions a generator $p$ for this ideal, but I don't understand what $p$ would be in this case, because isn't $\langle x,y \rangle$ alread the generator? I can mostly understand the rest of the proof, but I'm struggling to see what $p$ might look like, and why $x, y$ must be divisible by $p$. Is $p$ more accurately a polynomial that would be generated by $\langle x,y \rangle$? Or is this totally incorrect?
Any help would be really appreciated.
If $p$ is a generator, then $\langle x, y\rangle = \langle p\rangle$. Therefore in particular $x, y\in\langle p\rangle$, so that $x = pf$ and $y = pg$ for some $f, g\in R[x,y]$. Does that help?