how would you solve this? (generating sets ideals)

From: Carlos Ram?rez (solrac140_at_hotmail.com)
Date: 03/17/05 

Date: 17 Mar 2005 09:11:55 -0800

Hi, I had an exam and fortunately this was the only problem I wasn't
sure how to solve it.

Let I be the ideal in R[x,y] generated by the polynomial
x*(x^2+y^2-1),
and let J be the ideal in R[x,y] generated by the polynomial
y*(x^2+y^2 -1)
Find generators for I intersection J, I+J and I*J. (it doesn't have to
be
the minimal generating sets though). R= field of the real numbers.

I know that in the case of MONOMIALS then a generator for I
intersection J
would consist of taking the least common multiple of each pair and
then do the product, for I + J is simply the union of the elements of
I and J, and for IJ is the product.

The thing here is they are polynomials and we never saw an example
with them, although what I wrote is:
Let I = <x*(x^2+y^2-1)> = <x^3+ x*y^2 - x> , here is the step I think
is wrong:
      = <x^3, xy^2 , -x> = <x^3, xy^2, x>
That's the only thing I could came up with so it involves monomials
instead of polynomials, but probably is not true...
Then similarly for J = <yx^2, y^3, y>
And then work with them as monomials.

If this is incorrect then what was the way to solve it? (assuming you
only know that the fact about generating sets in the case of
monomials)
There must be something obvious which I wasn't able to see :/

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