Year: 2026
Type: article
Abstract: The product matrix of a finite commutative ring R={x1,x2,…,xn} and an element u∈R is the matrix Au(R)=[aij], where aij=1 if xixj=u, and aij=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 Au(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 Au(R) for a local ring R and any u∈R in two cases: when the Jacobson radical of R has either the maximal or the minimal possible index of nilpotency.
Source: Linear and Multilinear Algebra
Author: David Dolžan
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W7128488226