Locally compact models for approximate rings

Abstract By an approximate subring of a ring we mean additively symmetric subset X such that $$X\cdot \cup (X +X)$$ X · ∪ ( + ) is covered by finitely many additive translates . We prove each has locally compact model, i.e. homomorphism $$f :\langle \rangle \rightarrow S$$ f : ⟨ ⟩ → S for some S f [ ] relatively in and there neighborhood U 0 with $$f^{-1}[U] \subseteq 4X + \cdot 4X$$ - 1 [ U ] ⊆ 4 (where $$4X:=X+X+X+X$$ = ). This obtained as the quotient $$\langle $$ interpreted sufficiently saturated model its type-definable connected component. The main point to this component always exists. In order do that, extend basic theory model-theoretic components definable rings [developed Gismatullin et al. (J Symb Log First View: 1–35, 2022, https://doi.org/10.1017/jsl.2022.10 ) Krupiński (Ann Pure Appl Logic 173.7(July):103119, 2022) case generated subrings answer question from (2022) more general context subrings. Namely, let be (in structure M $$R:=\langle R Let $${\bar{X}}$$ ¯ interpretation elementary extension $${\bar{R}}:= \langle {\bar{X}} It follows Massicot Wagner Éc Polytech Math 2:55–63, 2015) exists smallest -type-definable subgroup $$({\bar{R}},+)$$ , bounded index, which denoted $$({\bar{R}},+)^{00}_M$$ M 00 $$({\bar{R}},+)^{00}_M {\bar{R}} ({\bar{R}},+)^{00}_M$$ two-sided ideal $${\bar{R}}$$ denote $${\bar{R}}^{00}_M$$ Then first sentence abstract just $${\bar{R}}/{\bar{R}}^{00}_M$$ / $$f: R {\bar{R}}/{\bar{R}}^{00}_M$$ map. fact, universal “definable” suitable sense) model. existence models can seen structural result about subrings: every recovered up commensurability preimage any should also have various applications get precise or even classification results. For example, paper, deduce [definable] positive characteristic commensurable contained $$4X easily implies given $$K,L \in \mathbb {N}$$ K L ∈ N constant C ( K , L -approximate (i.e. cover $$X (X+X)$$ $$\le L$$ ≤ )-commensurable Another application finite without zero divisors: $$K $$N(K) divisors either $$|X| <N(K)$$ | < $$K^{11}$$ 11 -commensurable