2024/03/14 by Ken A. Brown, Brown, Ken, Paula A. A. B. Carvalho +3
Mathematics · #16D50 #16P40 #16S35 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2403.09239
openalex publication_date 2024/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
This paper is a continuation of a project to determine which skew polynomial algebras S = R[θ; α] satisfy property (\diamond), namely that the injective hull of every simple S-module is locally artinian, where k is a field, R is a commutative noetherian k-algebra, and α is a k-algebra automorphism of R. Earlier work (which we review) and further analysis done here leads us to focus on the case where S is a primitive domain and R has Krull dimension 1 and contains an uncountable field. Then we show first that if |Spec(R)| is infinite then S does not satisfy (\diamond). Secondly we show that when R = k[X] and α(X) = qX where q ∈ k ∖ \0\ is not a root of unity then S does not satisfy (\diamond). This is in complete contrast to our earlier result that, when R = k[[X]] and α is an arbitrary k-algebra automorphism of infinite order, S satisfies (\diamond). A number of open questions are stated.