Eigenvalues of the product matrices of finite commutative rings

Abstract

The product matrix of a finite commutative ring $R=\{x_1,x_2,\dots ,x_n\}$ and an element $u\in R$ is the matrix $A_u(R)=[a_{ij}]$, where $a_{ij}=1$ if $x_i{x}_j=u$, and $a_{ij}=0$ otherwise. This provides a natural extension of the concept of the adjacency matrix of the zero-divisor graph of a ring, which has been studied extensively. In this paper, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ of odd order and a unit $u$. By studying the structure of a finite local ring, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ and any $u\in R$ in two cases: when the Jacobson radical of $R$ has either the maximal or the minimal possible index of nilpotency.