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