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arXiv · 1705.06596

Simple modules and their essential extensions for skew polynomial rings

Ken Brown·Paula A. A. B. Carvalho·Jerzy Matczuk
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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 $(\diamond)$, that for every simple $S$-module $V$ the injective hull $E_S(V)$ 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 $(\diamond)$ 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.

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Ken Brown, Paula A. A. B. Carvalho, Jerzy Matczuk. 2017-05-18. Simple modules and their essential extensions for skew polynomial rings. https://arxiv.org/abs/1705.06596

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