Locally compact models for approximate rings

Abstract

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