Showing two ideals are the same

[Juanriq]

Homework Statement

Let $I = (2x^2 + 3y^2 -11, x^2 -y^2 -3)$ and $J = (x^2 -4, y^2 -1)$. Show that these are the same.

Homework Equations

The Attempt at a Solution

Well, I thought that $I(x,y) = f_1(x,y) x + f_2(x,y)y$ does this mean that $f_1(x,y) = 2x^2 + 3y^2 -11$ for I? Am I supposed to add both components of I and then factor it into a (something)x + (something else )y? Thanks in advance, abstract algebra and I have a love-hate (mostly hate) relationship.

 

[micromass]

It suffices to show that the generators of I are in J, and that the generators in J are in I.

So first, we need to show that $$2x^2+3y^2-11$$ is in J (and analogous for the other generator). So we'll need to find polynomials $$P(x), Q(x)$$ such that

$$2x^2+3y^2-11=P(x)(x^2-4)+Q(x)(y^2-1)$$

Try to find these polynomials (hint: the polynomials are constants in this case)

 

[Juanriq]

That's it? P(x) = 2 and Q(x) = 3 for the first and M(x) = 1 and N(x) = -1 for the second? Thanks again micromass, you are a lifesaver.

 

[micromass]

Yes, but you're not done yet. You also need to show that the generators of J are in I. So do the same thing for $$x^2-4$$ and $$y^2-1$$...

 

[Juanriq]

Would have definitely forgot about that, grazie.