Proposition

The intersection $I \cap J$ of two principal ideals $I = \ev{f}, J = \ev{g} \subset k[x_1, \cdots, x_n]$ is a principal ideal.

Solution

Let $h = \lcm(f, g)$. Then $f \mid h$ and $g \mid h$, so $\ev{h} \subset \ev{f}$ and $\ev{h} \subset \ev{g}$. Thus $\ev{h} \subset \ev{f} \cap \ev{g}$.

Let $F \in \ev{f} \cap \ev{g}$. Thus $f$ and $g$ divide $F$. Since $h$ is the lowest common multiplier, $h \mid F$. Therefore, $F \in \ev{h}$.

Therefore, $\ev{f} \cap \ev{g} = \ev{h}$.