Injective hulls of simple modules over differential operator rings

TL;DR

This paper investigates the structure of injective hulls of simple modules over differential operator rings, identifying conditions for these modules to be locally Artinian and characterizing certain Ore extensions.

Contribution

It provides necessary conditions for injective hulls to be locally Artinian and characterizes specific Ore extensions with this property.

Findings

Injective hulls of simple modules are locally Artinian under certain conditions.

Characterization of Ore extensions where injective hulls are locally Artinian.

Conditions involving derivations and automorphisms for these properties.

Abstract

We study injective hulls of simple modules over differential operator rings $R[\theta; d]$, providing necessary conditions under which these modules are locally Artinian. As a consequence we characterize Ore extensions of $S=K[x][\theta;\sigma, d]$ for $d$ a $K$-linear derivation and $\sigma$ a $K$-linear automorphism of $K[x]$ such that injective hulls of simple $S$-modules are locally Artinian.