Simple modules and their essential extensions for skew polynomial rings

TL;DR

This paper investigates conditions under which skew polynomial rings over commutative Noetherian rings satisfy a property related to the Artinian nature of injective hulls of simple modules, providing complete characterizations in specific cases.

Contribution

It offers necessary and sufficient conditions for the property $(ullet)$ in primitive skew polynomial rings and a complete characterization when the base ring is an affine algebra over a field.

Findings

$(ullet)$ holds iff all simple modules are finite dimensional over $k$ for affine $k$-algebras.

Complete characterization for affine $k$-algebras with automorphism automorphisms.

Discussion of examples and open questions related to the property $(ullet)$.

Abstract

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.