Simple modules and their essential extensions for skew polynomial rings
| Autor: | Brown, Ken, Carvalho, Paula A. A. B., Matczuk, Jerzy |
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| Rok vydání: | 2017 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $R$ be a commutative Noetherian ring and $\alpha$ an automorphism of $R$. This paper addresses the question: when does the skew polynomial ring $S = R[\theta; \alpha]$ 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 $\alpha$ 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 $S = R[\theta;\alpha]$. The paper ends with a number of open questions. Comment: 23 pages; comments welcome |
| Databáze: | arXiv |
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