- $I_{1}=(x)$, 2. $I_{2}=(x,y^{2})$, 3. $I_{3}=(x-y,x+y)$, 4. $I_{4}=(x-y,x^{2}-y^{2})$.
My Attempt: It is easy to check that $I_{1}$ is prime but not maximal ideal.
For $I_{2}$, clearly $y^{2} \in (x,y^{2})$, but y,y does not belong to $(x,y^{2})$. THis implies $I_{2}$ is not prime ideal. Similarly, $I_{4}$ is not prime ideal.
$I_{3}$ is both prime and maximal ideal(not sure about $I_{3}$.
Your answers for $I_1,I_2,I_3$ are correct.
For $I_3$, it's easily verified that $(x-y,x+y)=(x,y)$, which is clearly maximal.
Your answer for $I_4$ is not correct, since $$I_4=(x-y,x^2-y^2)=(x-y)$$ which is prime (but not maximal).