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Let $R$ be a commutative Noetherian ring and $\alpha$ an automorphism of $R$.This paper addresses the question: when does the skew polynomial ring $S =R[\theta; \alpha]$ satisfy the property $(\diamond)$, that for every simple$S$-module $V$ the injective hull $E_S(V)$ of $V$ has all its finitelygenerated submodules Artinian. The question is largely reduced to the specialcase where $S$ is primitive, for which necessary and sufficient conditions arefound, which however do not between them cover all possibilities. Neverthelessa complete characterisation is found when $R$ is an affine algebra over a field$k$ and $\alpha$ is a $k$-algebra automorphism - in this case $(\diamond)$holds if and only if all simple $S$-modules are finite dimensional over $k$.This leads to a discussion, involving close study of some families of examples,of when this latter condition holds for affine $k$-algebras $S =R[\theta;\alpha]$. The paper ends with a number of open questions.

Simple modules and their essential extensions for skew polynomial rings