The smallest ideals of βN under addition and multiplication

It has been known for some time that the smallest ideals of (βN,+) and (βN,⋅) are disjoint and that the closure of the former meets the latter. We show here that the closure of the smallest ideal of (βN,⋅) misses the smallest ideal of (βN,+). Indeed, a stronger statement holds: it misses N∗+N∗.