The intersection and sum of irrelevant ideals are also irrelevant

Question

Definition: A homogenous ideal $I \subset K[x_0,\dots,x_n]$ is irrelevant if $\left <x_0^r,\dots,x_n^r \right> \subset I$ for some $r > 0$.

For $I \cap J$, this is probably circular logic, to argue that $\left <x_0^r,\dots,x_n^r \right> \subset I \cap J \subset I$.

This is to say if $\left <x_0^r,\dots,x_n^r \right> \subset I$ and $\left <x_0^u,\dots,x_n^u \right> \subset J.$

For $I + J$, can one just set the second coordinate to be $0$?

Answer 1

Suppose $I$ and $J$ are irrelevant; then $\langle x_0^r,\dots,x_n^r\rangle\subseteq I$ and $\langle x_0^s,\dots,x_n^s\rangle\subseteq J$ for some $r>0$ and $s>0$.

It's not restrictive to assume $r\le s$ (otherwise just switch the roles of $I$ and $J$). Since $x_i^s\in\langle x_0^r,\dots,x_n^r\rangle$, for $i=0,1,\dots,n$, we have $$ \langle x_0^s,\dots,x_n^s\rangle\subseteq \langle x_0^r,\dots,x_n^r\rangle\subseteq I $$ and so $$ \langle x_0^s,\dots,x_n^s\rangle\subseteq I\cap J $$ For the sum it's even easier: any ideal containing an irrelevant ideal is irrelevant as well.