Locally compact models for approximate rings

TL;DR

This paper establishes that approximate subrings in rings can be modeled by locally compact rings, leading to structural and classification results for approximate subrings in rings of positive characteristic and rings without zero divisors.

Contribution

It proves that approximate subrings have locally compact models and derives new structural and classification results for approximate subrings in specific ring classes.

Findings

Approximate subrings have locally compact models.

Approximate subrings in positive characteristic are additively commensurable with subrings.

Finite approximate subrings in rings without zero divisors are either small or close to subrings.

Abstract

By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. We prove that each approximate subring $X$ of a ring has a locally compact model, i.e. a ring homomorphism $f \colon \langle X \rangle \to S$ for some locally compact ring $S$ such that $f[X]$ is relatively compact in $S$ and there is a neighborhood $U$ of $0$ in $S$ with $f^{-1}[U] \subseteq 4X + X \cdot 4X$ (where $4X:=X+X+X+X$). This $S$ is obtained as the quotient of the ring $\langle X \rangle$ interpreted in a sufficiently saturated model by its type-definable ring connected component. The above theorem can be seen as a general structural result about approximate subrings: every approximate subring $X$ can be recovered up to additive commensurability as the preimage by a locally compact model $f \colon \langle X \rangle \to S$ of any relatively compact neighborhood of $0$ in $S$. It also leads to more precise structural or even classification results. For example, we deduce that every [definable] approximate subring $X$ of a ring of positive characteristic is additively commensurable with a [definable] subring contained in $4X + X \cdot 4X$. This implies that for any given $K,L \in \mathbb{N}$ there exists $C(K,L)$ such that every $K$-approximate subring $X$ (i.e. $K$ additive translates of $X$ cover $X \cdot X \cup (X+X)$) of a ring of positive characteristic $\leq L$ is additively $C(K,L)$-commensurable with a subring contained in $4X + X \cdot 4X$. We also deduce a classification of finite approximate subrings of rings without zero divisors: for every $K \in \mathbb{N}$ there exists $N(K) \in \mathbb{N}$ such that for every finite $K$-approximate subring $X$ of a ring without zero divisors either $|X| <N(K)$ or $4X + X \cdot 4X$ is a subring which is additively $K^{11}$-commensurable with $X$.