Eigenvalues of the product matrices of finite commutative rings

TL;DR

This paper investigates the eigenvalues of product matrices derived from finite commutative rings, extending the zero-divisor graph adjacency matrix concept, and provides explicit characteristic polynomials for specific local rings.

Contribution

It derives the characteristic polynomial of product matrices for finite local rings of odd order, considering cases based on the Jacobson radical's nilpotency index.

Findings

Characteristic polynomial formulas for local rings of odd order

Explicit eigenvalue structure for matrices over rings with specific radical properties

Extension of zero-divisor graph adjacency matrix analysis to product matrices

Abstract

The product matrix of a finite commutative ring $R=\{x_1,x_2,\ldots,x_n\}$ and an element $u \in R$ is the matrix $A_u(R)=[a_{ij}]$, where $a_{ij}=1$ if $x_ix_j=u$, 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.