Check if the following ideals in $F[x, y]$ are closed ideal

[Definition] Let $$F$$ be a field. For $$E \subseteq F^n$$. The ideal of $$E$$, denoted $$I(E)$$, is $$I(E)=\left\{ f \in F[x_1, \cdots, x_n] : f(x)=0 \ \ \forall x \in E \right\}.$$ An ideal $$I \subseteq F[x_1, \cdots x_n]$$ is a closed ideal if it is of the form $$I(E)$$ for some $$E \subseteq F^n$$

Determine if the following ideals are closed
(1) $$(xy,x+y)$$
(2) $$(y, x^2-1)$$

I have no idea how to check if a given ideal is closed when it has more than one generators. Can anyone help me?

• You need to check whether the implication $f(x,y)=0\implies f\in I$ for all common zeroes $(x,y)$ of the polynomials that generate $I$ holds or not. So, first determine the common zeroes in both cases. – Berci May 22 at 8:25

Let's look at $$(1)$$ for example. Since $$I(E)$$ is an ideal, it is sufficient to show that $$I(E) \subset (xy, x+y)$$ and $$xy, x+y \in I(E)$$. But if $$xy, x+y \in I(E)$$, we get that $$xy = x + y = 0$$. Since $$F$$ is a field, this only happens for $$x = y = 0$$, so $$E = \{(0,0)\}$$.
But $$I(\{(0,0)\} = F[x,y] \ne (xy, x+y)$$, so the ideal cannot be closed.