Height of Product of Ideals

Question

Let $R$ be any Noetherian ring and $I$ and $J$ be ideals in $R$. Is it possible to find an equation with $\mathrm{ht}(I)$ and $\mathrm{ht}(J)$ which give us $\mathrm{ht}(IJ)$?

Answer 1

If $P$ is minimal prime over $IJ$, then either $P$ is minimal over $I$ or $P$ is minimal over $J$. This follows from the fact that if $P$ is prime and $P$ contains $IJ$, then either $P$ contains $I$ or $P$ contains $J$ (it is a good exercise to try to prove that). As a consequence, $\operatorname{height}(IJ) = \min \left\{\operatorname{height}(I), \operatorname{height}(J)\right\}$.