I want to determine whether this ideals are contained in each other or not.
1) In $\mathbb{R}[x]$, $I_1= \langle x(1+x^2) \rangle$ and $I_2= \langle x^2(1+x) \rangle$. I can see that $I_1 \not\subseteq I_2$, since $I_2$ has $x$'s of degree 2 or higher, whereas in $I_1$ we have elements of degree 1. However, I don't know how to check that $I_2 \not\subseteq I_1$. Does anyone have any suggestion?
2) In $\mathbb{R}[x,y]$, consider $I_1= \langle (y-x^2)^2+(y-x^3)^4 \rangle$ and $I_2= \langle (y-x^2)^4 + (y-x^3)^2 \rangle$. In this one, I really don't know how to check that neither is contained in the other. Any hints or suggestions?
Thanks in advance!