On power sums of matrices over a finite commutative ring

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 $R=\mathbb{Z}\mathrm{/}n\mathbb{Z}$ 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 $d=2$, $\text{card}(R)\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, $1<k\equiv -1,0,1(\mathrm{m}\mathrm{o}\mathrm{d}6)$ and the only element $e\in R\setminus \{0\}$ such that $2e=0$ is idempotent, in which case the sum is $\text{diag}(e,e)$.