Let $a$ and $b$ are filters. The product $a\times b$ is defined as the filter (on the set of pairs) induced by the base $\{ A\times B | A\in a, B\in b \}$.

It is simple to show that product of a non-trivial ultrafilter with itself is not an ultrafilter (as it is not finer than the principal filter corresponding to the identity relation).

My question: Is product of every two (different) non-trivial ultrafilters always not an ultrafilter?

Note: *non-trivial ultrafilter* is the same as *non-principal ultrafilter*.