Determine which of the following ideals are prime, maximal ideal or neither in the polynomial ring $C[x,y]$

Question

$I_{1}=(x)$, 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 $I_{3}$.

I just want to know is my answer is correct? – User124356 Nov 29 '19 at 01:07 — User124356, Nov 29 '19 at 01:07

score 1 · Answer 1 · answered Nov 29 '19 at 01:12

1

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).

answered Nov 29 '19 at 01:12

quasi

58,772

Can I say $I_{4} \subset I_{3}$, then it is not maximal. – User124356 Nov 29 '19 at 01:16
Yes, it's a proper subset since, for example, $x\in I_3$ but $x\not\in I_4$. – quasi Nov 29 '19 at 01:18
Yeah. Thanks for your help. – User124356 Nov 29 '19 at 01:19

Determine which of the following ideals are prime, maximal ideal or neither in the polynomial ring $C[x,y]$

1 Answers1