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Let $J = (uw -v^2, u^3 - vw, w^3 -u^5) \subseteq \mathbb{C}[u,v,w]$ and let $I = (uw -v^2, u^3 - vw,w^2 - u^2v) \subseteq \mathbb{C}[u,v,w]$

Show $J \subset I$.

To me, it doesn't seem like this should be true? Help?

2 Answers 2

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Why not?

It suffices to show that every generator of $J$ is an element of $I$. This is obviously true for $uw-v^2$ and for $u^3-vw$.

So we just need to show that $w^3-u^5\in I$.

Note that $u^2(u^3-vw) = u^5 - u^2vw\in I$; and $w(w^2-u^2v) = w^3-u^2vw\in I$. Since $I$ is an ideal, $w(w^2-u^2v) - u^2(u^3-vw)\in I.$

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Hint $\ {\rm mod}\ I:\ \ (w^2)w\equiv u^2(vw)\equiv u^2(u^3)\:$ so every generator of $J$ is in $I$