Let R be a commutative Noetherian ring and an automorphism of R. This paper addresses the question: when does the skew polynomial ring S=R[;] satisfy the property (), that for every simple S-module V the injective hull ES(V) of V has all its finitely generated submodules Artinian. The question is largely reduced to the special case where S is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when R is an affine algebra over a field k and is a k-algebra automorphismin this case () 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[;]. The paper ends with a number of open questions.
Year of publication: 2019