Determinants of Matrices over Commutative Finite Principal Ideal Rings

TL;DR

This paper investigates the enumeration of invertible matrices over commutative finite principal ideal rings, generalizing known results from matrices over integers modulo m, and provides explicit formulas for fixed determinants.

Contribution

It derives formulas for counting matrices with a given determinant over finite principal ideal rings, extending previous work on integer modulo rings.

Findings

Number of matrices with fixed determinant over finite chain rings determined

Multiplicative property of matrix counts over product rings established

Generalization of integer modulo matrix results achieved

Abstract

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring ${R}$ of a fixed determinant $a$ is determined for all $a\in {R}$ and positive integers $n$. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo $m$.