  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.
