Pr\"ufer modules over Leavitt path algebras

TL;DR

This paper studies specific modules over Leavitt path algebras associated with finite graphs, classifies when these modules are injective, and describes their structure as injective hulls of simple modules.

Contribution

It introduces and analyzes a class of uniserial modules over Leavitt path algebras and classifies their injectivity based on properties of the underlying graph's closed paths.

Findings

Classified closed paths c for which U_{E,c-1} is injective.

Showed U_{E,c-1} is the injective hull of the Chen simple module V_{[c^∞]}.

Described the structure of these modules as uniserial, artinian, non-noetherian.

Abstract

Let $L_K(E)$ denote the Leavitt path algebra associated to the finite graph $E$ and field $K$. For any closed path $c$ in $E$, we define and investigate the uniserial, artinian, non-noetherian left $L_K(E)$-module $U_{E,c-1}$. The unique simple factor of each proper submodule of $U_{E,c-1}$ is isomorphic to the Chen simple module $V_{[c^\infty]}$. In our main result, we classify those closed paths $c$ for which $U_{E,c-1}$ is injective. In this situation, $U_{E,c-1}$ is the injective hull of $V_{[c^\infty]}$.