My question is rather simple and straightforward: I'm just looking for some good examples of coprime ideals in either $\mathbb{C}[x,y]$ or $\mathbb{C}[x,y,z]$ to be able to play around with. Could anyone please help me in that regard?
Look forward to what you can come up with.
Turns out that the question has a very simple answer. For any given polynomial $f \in \mathbb{C}[x,y,z]$, clearly the ideals $(f)$ and $(f+1)$ are coprime. No need to continue any further.