  If every subring of an integral domain is atomic, then we say that the latter
is hereditarily atomic. In this paper, we study hereditarily atomic domains.
First, we characterize when certain direct limits of Dedekind domains are
Dedekind domains in terms of atomic overrings. Then we use this
characterization to determine the fields that are hereditarily atomic. On the
other hand, we investigate hereditary atomicity in the context of rings of
polynomials and rings of Laurent polynomials, characterizing the fields and
rings whose rings of polynomials and rings of Laurent polynomials,
respectively, are hereditarily atomic. As a result, we obtain two classes of
hereditarily atomic domains that cannot be embedded into any hereditarily
atomic field. By contrast, we show that rings of power series are never
hereditarily atomic. Finally, we make some progress on the still open question
of whether every subring of a hereditarily atomic domain satisfies ACCP.
