Critical and injective modules over skew polynomial rings

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let & alpha; be a k-algebra automorphism of R, and define S to be the skew polynomial algebra R[& theta;; & alpha;]. We offer, under some additional assumptions on R, a criterion for S to have injective hulls of all simple S-modules locally Artinian -that is, for S to satisfy property (O). It is easy and well known that if & alpha; is of finite order, then S has this property, but in order to get the criterion when & alpha; has infinite order we found it necessary to classify all cyclic (Krull) critical S-modules in this case, a result which may be of independent interest. With the help of the above we show that S ⠂ = k[[X]][& theta;, & alpha;] satisfies (o) for all k-algebra automorphisms & alpha; of k[[X]].& COPY; 2023 Elsevier B.V. All rights reserved.