On power sums of matrices over a finite commutative ring

TL;DR

This paper investigates the sum of the k-th powers of all matrices over finite commutative rings, providing complete solutions for certain cases and conjecturing a general pattern based on computational evidence.

Contribution

It completely solves the problem for R=Z/nZ and offers partial results for general finite rings, proposing a conjecture on the sum's behavior.

Findings

Sum is zero in most cases for matrix rings over finite rings.

Explicit sum formulas are given for R=Z/nZ.

Conjecture: sum is non-zero only under specific algebraic conditions.

Abstract

In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $\mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case $R=\mathbb{Z}/n\mathbb{Z}$ and give some results that compute the value of this sum if $R$ is an arbitrary finite commutative ring $R$ for many values of $k$ and $d$. Finally, based on computational evidence and using some technical results proved in the paper we conjecture that the sum of the $k$-th powers of all the elements of the matrix ring $\mathbb{M}_d(R)$ is always $0$ unless $d=2$, $\textrm{card}(R) \equiv 2 \pmod 4$, $1<k\equiv -1,0,1 \pmod 6$ and the only element $e\in R \setminus \{0\}$ such that $2e =0$ is idempotent, in which case the sum is $\textrm{diag}(e,e)$.