TITLE: Existence of a specific type of ultrafilter
QUESTION [7 upvotes]: Does there exist an ultrafilter $\omega$ on $N$ with the following property:
$\forall A \in \omega$ we have that $kA \in \omega$ for a fixed $k \in N$. Where $kA = \{ka : a \in A\}$.
I do not believe there is such an ultrafilter, but could not immediately see why one does not exist. An answer that shows there is one would be highly welcomed. Thanks for the any help.

REPLY [6 votes]: Assuming that $k>1$, and that you do not care about the principal ultrafilter generated by $\{0\}$, there are no such ultrafilters: Fix $k>1$, and consider an ultrafilter $\mathcal U$ on $\mathbb N$ as described. Note that $\mathcal U$ is non-principal (else $\mathcal U$ contains a singleton $\{s\}$, which is disjoint from $\{ks\}$ unless $s=0$, which we are explicitly excluding).  
Given $A\in\mathcal U$, we may assume that $0\notin A$ since $\mathcal U$ is non-principal. Now let $A_E$ be the subset of $A$ consisting of those elements of $A$ such that the largest power of $k$ dividing them is even, and let $A_O=A\setminus A_E$. Precisely one of these two sets, call it $B$, is in $\mathcal U$. Now note that $kB$ is disjoint from $B$, and we are done.