Eigenvalues of the product matrices of finite commutative rings
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.
Fields of Study & Subjects
Rings
Modules
and Algebras
Graph theory and applications
Matrix Theory and Algorithms
Publication details
- Journal / source
- Linear and Multilinear Algebra
- Volume
- 74
- Issue
- 4
- Pages
- 449–461
- Publication date
- 10 February 2026
- Language
- EN