The number of atoms in a primefree atomic domain

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 n Z+ with n 3 and 4 m 1173 there is an atomic domain with precisely atoms, precisely m maximal ideals and no prime elements. The Proofs involve aninterplay of commutative algebra, algebraic number theory and additive number theory.