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Preprint Number 2145

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2145. Krzysztof Krupiński
Locally compact models for approximate rings

Submission date: 10 March 2022


Comments: In all the citations of [GJK20], we referred to the numbering of the results from the version of that paper which will be published in the Journal of Symbolic Logic. The numbering of sections in the version of [GJK20] which is on arXiv should be increased by 1 to get the numbering to which we refer, so each result numbered as n.m in the version on arXiv is referred to as n+1.m in this paper.

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 : ⟨ 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 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 [GJK20] and [KR20]) to the case of rings generated by definable approximate subgrings and we answer a question from [KR20] 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:=⟨ X ⟩. Let X̅ be the interpretation of X in a sufficiently saturated elementary extension and R̅ := ⟨ X̅ ⟩. It follows from [MW15] that there exists the smallest M-type-definable subgroup of (R̅,+) of bounded index, which is denoted by (R̅,+)^{00}_M. We prove that (R̅,+)^{00}_M + R̅ . (R̅,+)^{00}_M is the smallest M-type-definable two-sided ideal of R̅ of bounded index, which we denote by R̅^{00}_M. Then S in the first sentence of the abstract is just R̅/R̅^{00}_M and f: R → R̅/R̅^{00}_M is the quotient map. In fact, f is the universal “definable” (in a suitable sense) locally compact model.

Mathematics Subject Classification: 03C60, 03C98, 11B30, 11P70, 16B70, 17A01, 20A15, 20N99

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Full text arXiv 2203.05609: pdf, ps.

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