  This paper is a continuation of a project to determine which skew polynomial
algebras $S = R[\theta; \alpha]$ satisfy property $(\diamond)$, namely that the
injective hull of every simple $S$-module is locally artinian, where $k$ is a
field, $R$ is a commutative noetherian $k$-algebra, and $\alpha$ is a
$k$-algebra automorphism of $R$. Earlier work (which we review) and further
analysis done here leads us to focus on the case where $S$ is a primitive
domain and $R$ has Krull dimension 1 and contains an uncountable field. Then we
show first that if $|\mathrm{Spec}(R)|$ is infinite then $S$ does not satisfy
$(\diamond)$. Secondly we show that when $R = k[X]_{<X>}$ and $\alpha (X) = qX$
where $q \in k \setminus \{0\}$ is not a root of unity then $S$ does not
satisfy $(\diamond)$. This is in complete contrast to our earlier result that,
when $R = k[[X]]$ and $\alpha$ is an arbitrary $k$-algebra automorphism of
infinite order, $S$ satisfies $(\diamond)$. A number of open questions are
stated.
