The number of atoms in a primefree atomic domain. Issue 12 (2nd December 2017)

• Record Type: Journal Article
• Authors: Clark, Pete L.; Gosavi, Saurabh; Pollack, Paul
• Is Part Of: Communications in algebra. Volume 45:Issue 12(2017)
• Page Start: 5431
• Page End: 5442
• Publication Date: 2017-12-02
• Subjects: Additive prime number theory -- atomic domain -- Cohen–Kaplansky domain; Primary: 13F15 -- Secondary: 11R27, 11P32; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1310877
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: 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 and4 ≤ m ≤ n 3 there is an atomic domain with precisely n atoms, precisely m maximal ideals and no prime elements. The proofs involve an interplay of commutative algebra, algebraic number theory and additive number theory.