My Math Forum the zero set of an ideal

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 January 28th, 2014, 06:24 AM #1 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 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?
 January 28th, 2014, 11:35 PM #2 Senior Member   Joined: Mar 2012 Posts: 294 Thanks: 88 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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### zero set of an ideal

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