I am a graduate student.I am trying to solve this problem: How to show that an ideal is maximal.
In this problem I am given the ring $\mathbb Q[x,y]$ and the ideal $I=(x-1,y^2+2)$ and I am asked to find whether this ideal is maximal.One way to show maximality is that the quotient ring is a field.The solution given in the above link follows that,as expected.But then they claim that first we can quotient out by $(x-1)$ and then by $(y^2+2)$.This is the point which I do not get.Why is it true that $\mathbb Q[x,y]/(x-1,y^2+2)\simeq \frac{\mathbb Q[x,y]/(x-1)}{(y^2+2)}$.The rest of the solution is clear to me that $\mathbb Q[x,y]/(x-1)$ is equal to $\mathbb Q[y]$ and we are trying to calculate $\mathbb Q[y]/(y^2+2)$ which is essentially $\mathbb Q[i\sqrt 2]$ which is a field.Please help me resolve this doubt.Does it follow from any theorem?
This is the ideal correspondence theorem. That is, let $R$ be a ring and $I\subset R$ an ideal. Ideals of $R/I$ correspond to ideals $J\subset R$ with $I\subset J$. Moreover, if $J'=\{x+I|x\in J\}$ is the ideal of $R/I$ corresponding to $J$, then
$$(R/I)/J'\cong R/J$$
In this particular case, $R=\mathbb{Q}[x,y]$, $I=(x-1)$, $J=(x-1,y^2+2)$.