The Artin--Rees lemma and size of spaces over nonassociative complete filtered rings

Abstract

This paper studies nonassociative filtered rings using associated gradations. We show that a complete filtered ring $R$ with affine associated graded ring generated in degree $1$ is a local ring, and prove that the Artin--Rees lemma holds for $R$. Assuming finiteness of the residue field, we derive asymptotics for abelian groups with an operation of $R$, and identify classes of torsion elements for spaces over a central extension of $R$.