  An integral domain is called atomic if every nonzero nonunit element factors
into irreducibles. On the other hand, an integral domain is said to satisfy the
ascending chain condition on principal ideals (ACCP) if every ascending chain
of principal ideals terminates. It was asserted by Cohn back in the sixties
that every atomic domain satisfies the ACCP, but such an assertion was refuted
by Grams in the seventies with an explicit construction of a neat example.
Still, atomic domains without the ACCP are notoriously elusive, and just a few
classes have been found since Grams' first construction. In the first part of
this paper, we generalize Grams' construction to provide new classes of atomic
domains without the ACCP. In the second part of this paper, we construct what
seems to be the first atomic semigroup ring without the ACCP in the existing
literature.
