On power sums of matrices over a finite commutative ring

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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 Md(R) with d>1 and R a finite commutative ring. We completely solve the problem in the case R=Z/nZ 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 Md(R) is always 0 unless d=2, card(R) ≡ 2 4, 1<k≡ -1,0,1 6 and the only element e∈ R {0} such that 2e =0 is idempotent, in which case the sum is diag(e,e).