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Simple modules and their essential extensions for skew polynomial rings

Preprint

<p>Let <em>R</em> be a commutative Noetherian ring and <em>α</em> an automorphism of <em>R</em>. This paper addresses the question: when does the skew polynomial ring <em>S</em>=<em>R</em>[<em>θ</em>;<em>α</em>] satisfy the property (⋄), that for every simple <em>S</em>-module <em>V</em> the injective hull <em>E</em><em>S</em>(<em>V</em>) of <em>V</em> has all its finitely generated submodules Artinian. The question is largely reduced to the special case where <em>S</em> 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 <em>R</em> is an affine algebra over a field <em>k</em> and <em>α</em> is a <em>k</em>-algebra automorphism - in this case (⋄) holds if and only if all simple <em>S</em>-modules are finite dimensional over <em>k</em>. This leads to a discussion, involving close study of some families of examples, of when this latter condition holds for affine <em>k</em>-algebras <em>S</em>=<em>R</em>[<em>θ</em>;<em>α</em>]. The paper ends with a number of open questions.</p>

Paula Carvalho

Jerzy Matczuk

Ken Brown

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Year of publication: 2017

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