# Projective indecomposable modules and quivers for monoid algebras

## TL;DR

This paper constructs projective indecomposable modules and describes the quiver for a broad class of monoid algebras, extending previous results to include many finite monoid families.

## Contribution

It provides a unified construction and description of modules and quivers for monoid algebras, covering new classes of monoids not previously analyzed.

## Key findings

- Constructed projective indecomposable modules for the class of monoid algebras.
- Described the quiver structure for these monoid algebras.
- Extended known results to include all finite monoids with at most one idempotent generator per principal right ideal.

## Abstract

We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, $\mathscr J$-trivial and $\mathscr R$-trivial monoids and left regular bands of groups.