A Carlitz–von Staudt type theorem for finite rings

Abstract

We compute the kth power-sums (for all k>0) over an arbitrary finite unital ring R. This unifies and extends the work of Brawley et al. (1974) [1] for matrix rings, with folklore results for finite fields and finite cyclic groups, and more general recent results of Grau and Oller-Marcén (2017) [12] for commutative rings. As an application, we resolve a conjecture by Fortuny Ayuso et al. (2017) [7] on zeta values for matrix rings over finite commutative rings. We further recast our main result via zeta values over polynomial rings, and end by classifying the translation-invariant polynomials over a large class of finite commutative rings.