We consider the structure of modules over Noetherian rings. A Noetherian ring S whose simple modules have the property that their finitely generated essential extensions are Artinian is said to satisfy property (⋄). For commutative Noetherian rings the validity of (⋄) is due to Matlis (1958).
In this talk we will discuss (⋄) for skew polynomial rings S = R[θ; α] where R is a commutative Noetherian ring and α is an automorphism of R, with the indeterminate θ satisfying the relation θr = α(r)θ for all r ∈ R.
A complete characterization is found when R is an affine algebra over a field K and α is a K-automorphism. We will discuss in some detail the case when R = C[x, y] and α is a C-algebra automorphism and indicate some open questions.

This talk is based on a paper with Ken Brown and Jerzy Matczuk.