The quiver of an affine monoid

TL;DR

This paper computes the Gabriel quiver of the complex algebra of the affine monoid associated with any finite commutative ring, linking algebraic structures with representation theory.

Contribution

It provides an explicit computation of the Gabriel quiver for the complex algebra of affine monoids over finite commutative rings, a previously unaddressed problem.

Findings

Explicit Gabriel quiver for affine monoids over finite rings

Connection between affine monoid structure and representation theory

Extension of known results to all finite commutative rings

Abstract

If $R$ is a finite commutative ring, then the affine monoid of $R$ is the monoid of all affine mappings $x\mapsto ax+b$ on $R$. Alternatively, it is the semidirect product of the multiplicative monoid of $R$ with the additive group of $R$. In this paper we compute the Gabriel quiver of the complex algebra of the affine monoid of any finite commutative ring.