Simple modules and their essential extensions for skew polynomial rings

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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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
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
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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