The number of atoms in an atomic domain

الأصل · EN

We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all m,n ∈ Z+ with n ≥ 3 and 4 ≤ m ≤ n/3 there is an atomic domain with precisely n atoms, precisely m maximal ideals and no prime elements. The proofs use both commutative algebra and additive number theory.