Abstract
In this paper we compute the sum of the k-th powers of all the elements of a finite commutative unital ring, thus generalizing known results for finite fields, the rings of integers modulo n or the ring of Gaussian integers modulo n. As an application, we focus on quotient rings of the form (Z/nZ)[x]/(f(x)) for a polynomial f∈Z[x].
Highlights
For a finite ring R and k ≥ 1, we define the power sum Sk(R) := rk. r∈RJ.M
Throughout the paper we will deal only with finite commutative unital rings and our main objective will be the computation of Sk(R) in this case
We begin this section with the following straightforward result
Summary
For a finite ring R and k ≥ 1, we define the power sum Sk(R) := rk. J.M. Throughout the paper we will deal only with finite commutative unital rings and our main objective will be the computation of Sk(R) in this case. The problem of computing Sk(R) has been completely solved only for some particular families of finite rings. Let R be a finite commutative unital ring and assume that |R| = ps11 · · · psl l. This implies that char(R) = pt11 · · · ptll with 1 ≤ ti ≤ si for every i. We completely solve the problem of computing Sk(R) for any finite commutative unital ring.
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