Hereditary atomicity and ACCP in abelian groups

Abstract

A cancellative and commutative monoid $M$ is atomic if every non-invertible element of $M$ factors into irreducibles (also called atoms), and $M$ is hereditarily atomic if every submonoid of $M$ is atomic. In addition, $M$ is hereditary ACCP if every submonoid of $M$ satisfies the ascending chain condition on principal ideals (ACCP). Our primary purpose in this paper is to determine which abelian groups are hereditarily atomic. In doing so, we discover that in the class of abelian groups the properties of being hereditarily atomic and being hereditary ACCP are equivalent. Once we have determined the abelian groups that are hereditarily atomic, we will use this knowledge to determine the commutative group algebras that are hereditarily atomic, that is, the commutative group algebras satisfying that all their subrings are atomic. The interplay between atomicity and the ACCP is a subject of current active investigation. Throughout our journey, we will discuss several examples connecting (hereditary) atomicity and the ACCP, including, for each integer $d$ with $d \ge 2$, a construction of a rank-$d$ additive submonoid of $\mathbb{Z}^d$ that is atomic but does not satisfy the ACCP.