Critical and injective modules over skew polynomial rings

Let R be a commutative local k-algebra of Krull dimension one, where k is field. α automorphism R, and define S to the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on criterion for have injective hulls all simple S-modules locally Artinian - that is, satisfy property (⋄). It easy well known if finite order, then has this property, but in order get when infinite we found it necessary classify cyclic (Krull) critical case, result which may independent interest. With help above show Sˆ=k[[X]][θ,α] satisfies (⋄) automorphisms k[[X]].