Assuming that $I,J\subset S$ are monomial ideals, is it true that $I+J$ is monomial? What about $IJ$ and $I\cap J$?

Question

Let $\mathbb{k}$ be a field and $S=\mathbb{k}[x_1,\dots,x_k]$ be a ring of polynomials over $\mathbb{k}$.

Assuming that ideals $I,J\subset S$ are monomial, is it true that $I+J$ is monomial? What about $IJ$ and $I\cap J$?

I do not know how to approach this problem. If we take for example $I=(x_1,x_2)$ and $I=(x_2,x_3)$ then how to find $I+J$ and so on?

Answer 1

To answer this it is useful to think of ideals in terms of generating sets. You know what this is already because monomial ideal is defined in terms of generators.

Now $I+J$ is another way of saying "the ideal generated by $I$ and $J$". Perhaps prove this first. Then if $I$ and $J$ are generated by some subsets $I_0$ and $J_0$, what could you use as generators for $I+J$? For $I\cdot J$?

Answer 2

$I\cap J$ is a monomial ideal

We use a result which is stated that $I$ is a monomial ideal then for any $f\in I$, one has $supp(f)\subseteq I$.

Let $f\in I\cap J$ as both $I$ and $J$ are monomial ideals implies $supp(f)\subseteq I$ and $supp(f)\subseteq J $. which implies $supp(f)\subseteq I\cap J$.

Hence $I\cap J$ is monomial ideal