Show by counter example that $\{xy : x ∈ I,y ∈ J\}$ need not be an ideal

Question

On this page I find an example saying that $x^2+y^2$ is not a product of two elements in the set $(x, y) = \{ax + by : a, b ∈ \mathbb R\}$.

But so far I think we can take $x^2+y^2∈ I$ and $1∈ J$ and thus it is a product of $u ∈ I,v ∈ J$.

Is it wrong? Thanks for explanation.

Answer 1

In the example given, we have $I=J=\left\{ ax+by:a,b\in\mathbb{R}\right\}$. But we can not have $x^{2}+y^{2}\in I$ because $I$ consists only of polynomials of the form $ax+by$ for any real $a$ and $b$.

EDIT: As another user has pointed out, there is a typo in the pfd file. The $a$ and $b$ are supposed to be elements of $\mathbb{R}[x,y]$. In this case, it is not true that $1\in I$. Try and see why.