A Class of Atomic Rings

Abstract

Let R be a commutative ring with identity. A nonunit element a ∈ R is an atom if a = bc (b, c ∈ R) implies (a) = (b) or (a) = (c), and R is atomic if each nonunit of R is a finite product of atoms. A ring R is a (weak) CK ring if R is atomic and (each maximal ideal of) R contains only finitely many nonassociate atoms. We show that the following are equivalent: (1) R is a weak CK ring, (2) each (prime) ideal of R is a finite union of principal ideals, (3) R is atomic and each maximal ideal is a finite union of principal ideals, and (4) R is a finite direct product of finite local rings, special principal ideal rings (SPIRs), and (one-dimensional) Noetherian domains in which every maximal ideal is a finite union of principal ideals (or equivalently, weak CK domains). We show that a domain R is a weak CK domain if and only if R is a Noetherian domain with R M a CK domain for each maximal ideal M of R and Pic(R) = 0.