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 $R$. Ayyer and Steinberg studied Markov chains coming from random applications of $ax+b$ mappings on $R$ using the representation theory of the affine monoid $Aff(R)$ of all such maps. They described the simple $\mathbb{C}Aff(R)$-modules. In this work, we compute the Gabriel quiver of the complex algebra of the affine monoid of any finite commutative ring. This is joint work with B. Steinberg.