A Carlitz-von Staudt type theorem for finite rings

Abstract

We compute the $k$th power-sums (for all $k>0$) over an arbitrary finite unital ring $R$. This unifies and extends the work of Brawley, Carlitz, and Levine for matrix rings [Duke Math. J. 1974], with folklore results for finite fields and finite cyclic groups, and more general recent results of Grau and Oller-Marcen for commutative rings [Finite Fields Appl. 2017]. As an application, we resolve a conjecture by Fortuny Ayuso, Grau, Oller-Marcen, and Rua on zeta values for matrix rings over finite commutative rings [Internat. J. Algebra Comput. 2017]. 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.

18 pages. Final version, to appear in Linear Algebra and its Applications (the journal version is longer)

References in corpus (2)

• On the congruence $1^m+2^m+\cdots +m^m\equiv n(\mathrm{mod}m)$ with $n|m$
• On the center of the ring of differential operators on a smooth variety over $\bZ/p^n\bZ$