The smallest ideals of under addition and multiplication

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