2022/03/10 by Krzysztof Krupiński, Krupiński, Krzysztof · 1 citation
Mathematics · #Advanced Topology and Set Theory #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2203.05609
By an approximate subring of a ring we mean an additively symmetric subset X such that 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 \colon ⟨ 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] ⊆ 4X + X ⋅ 4X (where 4X:=X+X+X+X). This S is obtained as the quotient of the ring ⟨ X ⟩ 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 ⟨ X ⟩ → 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 ⋅ 4X. This implies that for any given K,L ∈ ℕ there exists C(K,L) such that every K-approximate subring X (i.e. K additive translates of X cover X ⋅ X ∪ (X+X)) of a ring of positive characteristic ≤ L is additively C(K,L)-commensurable with a subring contained in 4X + X ⋅ 4X. We also deduce a classification of finite approximate subrings of rings without zero divisors: for every K ∈ ℕ there exists N(K) ∈ ℕ such that for every finite K-approximate subring X of a ring without zero divisors either |X|