Brown, K. , Carvalho, P. A.A.B. and Matczuk, J. (2019) Simple modules and their essential extensions for skew polynomial rings. Mathematische Zeitschrift, 291(3-4), pp. 877-903. (doi: 10.1007/s00209-018-2128-8)
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Abstract
Let R be a commutative Noetherian ring and α an automorphism of R. This paper addresses the question: when does the skew polynomial ring S=R[θ;α] satisfy the property (⋄) , that for every simple S-module V the injective hull ES(V) of V has all its finitely generated submodules Artinian. The question is largely reduced to the special case where S is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when R is an affine algebra over a field k and α is a k-algebra automorphism—in this case (⋄) holds if and only if all simple S-modules are finite dimensional over k. This leads to a discussion, involving close study of some families of examples, of when this latter condition holds for affine k-algebras S=R[θ;α] . The paper ends with a number of open questions.
Item Type: | Articles |
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Status: | Published |
Refereed: | Yes |
Glasgow Author(s) Enlighten ID: | Brown, Professor Ken |
Authors: | Brown, K., Carvalho, P. A.A.B., and Matczuk, J. |
College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
Journal Name: | Mathematische Zeitschrift |
Publisher: | Springer |
ISSN: | 0025-5874 |
ISSN (Online): | 1432-1823 |
Published Online: | 21 August 2018 |
Copyright Holders: | Copyright © 2018 The Authors |
First Published: | First published in Mathematische Zeitschrift 291:877–903 |
Publisher Policy: | Reproduced under a Creative Commons License |
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