I have to answer to questions. I try to give the answer myself and would appreciate if you told me if they are correct.
In the ring $\mathbb{C}[x,y]$, do the following inclusions of ideals hold?
- $(x-1,y-2)\subset(y^2-4x)$
- $(y^2-4x)\subset(x-1,y-2)$
I start with 2.: Yes, this inclusion holds, since the ideal $m:=(x-1,y-2)$ is a maximal ideal in the ring $\mathbb{C}[x,y]$. Therefore, the ideal $(y^2-4x)$ is contained in $m$ since $y^2-4x$ vanishes for $x=1$ and $y=2$.
Now 1.: I would say that the inclusion does not hold. If this was the case, then we would have $x-1=(y^2-4x)\cdot p(x,y)$ for some polynomial $p(x,y)\in\mathbb{C}[x,y]$ (same reasoning for $y-2$). But there does not exist such a $p(x,y)$ and we have a contradiction.
An additional question with regard to 1.: How do I prove that $m$ is indeed maximal in $\mathbb{C}[x,y]$?
Thank you very much for your help.
