I know the following simple fact is true, but I can't find a good proof:
Over the naturals, the only ultrafilter $\mathcal U$ such that $\mathcal U \oplus \mathcal U = \mathcal U \odot \mathcal U$ is the principal ultrafilter $2$.
Thanks!
(sum and product are defined as:
$A \in \mathcal U \oplus \mathcal V \Leftrightarrow \{n \,|\, A - n \in \mathcal V \} \in \mathcal U$
$A \in \mathcal U \odot \mathcal V \Leftrightarrow \{n \,|\, A/n \in \mathcal V \} \in \mathcal U$
where $A-n=\{m \, |\, m+n \in A\}$ and $A/n=\{m \, | \, m \cdot n \in A \}$ )