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-   -   the zero set of an ideal (http://mymathforum.com/abstract-algebra/41084-zero-set-ideal.html)

 rayman January 28th, 2014 06:24 AM

the zero set of an ideal

Given $J=(XY,XZ,YZ)$
Find the zero set $V(J)$

By definition : $V(J):=\{x\in \mathbb{A}^{n}\;\;|f(x)=0,\;\;\forall f\in J\}$

So I show that we must have $\begin{cases} XY=0 \\ YZ=0\\XZ=0 \end{cases}$

and from this

$\begin{cases}X=0 \vee Y=0 \\Y=0 \vee Z=0\\Y=0 \vee Z=0\end{cases}$

hence $V(J)=\{(0,0,0)\subset \mathbb{A}^{3}\}$ right?

 Deveno January 28th, 2014 11:35 PM

Re: the zero set of an ideal

It seems to me your zero-set is too small. For example, what about (0,0,c), (0,b,0) or (a,0,0)?

Surely every element of J annihilates all of those....

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