On cyclic essential extensions of simple modules over differential operator rings

TL;DR

This paper investigates conditions under which cyclic essential extensions of simple modules over differential operator rings are Artinian, focusing on cases where the base ring is d-simple, d-primitive, or an affine algebra of Kull dimension 2.

Contribution

It provides new criteria for Artinian properties of cyclic essential extensions over differential operator rings, especially in specific algebraic settings.

Findings

Cyclic essential extensions are Artinian under certain conditions.

Characterization of differential operator rings C[x,y][z;d] with Artinian cyclic extensions.

Results for rings where R is d-simple or d-primitive.

Abstract

In this paper we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[z;d] are Artinian. In particular, we study the case when R is either d-simple or d-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application we characterize the differential operator rings C[x,y][z;d] for which cyclic essential extensions of simple modules are Artinian.