Property () for Ore extensions of small Krull dimension

This paper is a continuation of a project to determine which skew polynomial algebras S = R[θ; α] satisfy property (), 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 α 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 |Spec(R)| is infinite then S does not satisfy (). Secondly we show that when R = k[X]<X> and α (X) = qX where q ∈ k \0\ is not a root of unity then S does not satisfy (). This is in complete contrast to our earlier result that, when $R = k[[X]]$ and α is an arbitrary k-algebra automorphism of infinite order, S satisfies (). A number of open questions are stated.