Kronecker modules generated by modules of length 2

TL;DR

This paper investigates modules generated by length-2 modules, called bristled modules, focusing on Kronecker modules, and demonstrates their abundance for Kronecker quivers with three or more arrows.

Contribution

It introduces and studies bristled modules generated by length-2 modules in the context of Kronecker quivers, revealing their plentiful existence for n ≥ 3.

Findings

Bristled Kronecker modules are abundant for n ≥ 3.

Modules generated by length-2 modules are less explored and studied.

The paper provides foundational results on the structure of these modules.

Abstract

Let $\Lambda$ be a ring and $\mathcal N$ a class of $\Lambda$-modules. A $\Lambda$-module is said to be generated by $\mathcal N$ provided that it is a factor module of a direct sum of modules in $\mathcal N$. The semi-simple $\Lambda$-modules are just the $\Lambda$-modules which are generated by the $\Lambda$-modules of length 1. It seems that the modules which are generated by the modules of length $2$ (we call them bristled modules) have not attracted the interest they deserve. In this paper we deal with the basic case of the Kronecker modules, these are the (finite-dimensional) representations of an $n$-Kronecker quiver, where $n$ is a natural number. We show that for $n\ge 3$, there is an abundance of bristled Kronecker modules.