Critical and injective modules over skew polynomial rings

Brown, K. , Carvalho, P. A.A.B. and Matczuk, J. (2023) Critical and injective modules over skew polynomial rings. Journal of Pure and Applied Algebra, 227(11), 107441. (doi: 10.1016/j.jpaa.2023.107441)

Abstract

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let α be a k-algebra automorphism of R, and define S to be the skew polynomial algebra R [ θ ; α ] . 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 ( ⋄ ) . It is easy and well known that if α is of finite order, then S has this property, but in order to get the criterion when α 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 ˆ = k [ [ X ] ] [ θ , α ] satisfies ( ⋄ ) for all k-algebra automorphisms α of k [ [ X ] ] .

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Brown, Professor Ken and Carvalho, Dr Paula
• Authors: Brown, K., Carvalho, P. A.A.B., and Matczuk, J.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Journal of Pure and Applied Algebra
• Publisher: Elsevier
• ISSN: 0022-4049
• ISSN (Online): 1873-1376
• Published Online: 22 May 2023
• Copyright Holders: Copyright © 2023 Elsevier B.V.
• First Published: First published in Journal of Pure and Applied Algebra 227(11):107441
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
300872		Aspects of noncommutative geometry and noncommutative algebra	Kenneth Brown	Leverhulme Trust (LEVERHUL)	EM-2017-081\9	M&S - Mathematics