On noncommutative bounded factorization domains and prime rings

Hereditary atomicity in integral domains

“…Further constructions have been crafted recently: in [17], Boynton and the first author constructed an atomic pullback ring that does not satisfy the ACCP, while a more thorough investigation of the class of atomic domains not satisfying the ACCP was carried out by Li and the first author in [48], where further constructions of atomic domains not satisfying the ACCP were provided. Finally, an example of a non-commutative atomic ring not satisfying the ACCP was recently constructed by Bell et al in [16].…”

A characterisation of atomicity

“…More recently, J. G. Boynton and J. Coykendall [ 2 ] have used pullbacks of commutative rings to generate large families of atomic commutative domains that do not satisfy the ACCP; F. Gotti and B. Li [ 19 , theorem 4·4] have built what appears to be the first example of an atomic, commutative monoid domain without the ACCP; and J. Bell et al [ 1 , proposition 7·6] have provided the first example of an atomic, non-commutative, finitely presented monoid domain that satisfies neither the ACCPL nor the ACCPR (see also [ 13 ] for some related results on monoid rings, atomicity, and the ACCP).…”