The following question is about understanding a basic definition. the example involved is simple.
According to an answer to another question on this site, The set generated by $x,y$ is an ideal in $\mathbb{Q}[x,y]$. I can't understand why it is an ideal. In what way does it absorb other elements of $\mathbb{Q}[x,y]$ (by multiplication operation). Maybe the question is what elements does $(x,y)$ contain, since I know what $\mathbb{Q}[x,y]$ means and what polynomial multiplication means.
Thanks.
As Crostul mentions in his comment, the ideal $(x, y)$ by definition consists of the elements of the form $$x f(x, y) + y g(x, y)$$ (but NB this representation need not be unique).
Now, any element of $\Bbb Q[x, y]$ can be uniquely written as $$a_{00} + \sum_{i, j > 0} a_{ij} x^i y^j$$ (for all but finitely many $a_{ij}$ nonzero), so $p(x, y) \in \Bbb Q[x, y]$ is in the ideal $(x, y)$ iff the constant term $a_{00}$ is zero. On the other hand, $a_{00} = p(0, 0)$, so $(x, y)$ consists precisely of the polynomials that vanish at the origin.