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[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?

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  • $\begingroup$ 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. $\endgroup$ – Berci May 22 at 8:25
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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.

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