Locally compact models for approximate rings

Abstract

By an approximate subring of a ring we mean an additively symmetric subset X such that X ¨X Y pX `Xq 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 : xXy Ñ S for some locally compact ring S such that f rXs is relatively compact in S and there is a neighborhood U of 0 in S with f ´1rU s Ď 4X `X ¨4X (where 4X :" X `X `X `X). This S is obtained as the quotient of the ring xXy 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 :" xXy. Let X be the interpretation of X in a sufficiently saturated elementary extension and R :" x Xy. It follows from [MW15] that there exists the smallest M -type-definable subgroup of p R, `q of bounded index, which is denoted by p R, `q00M . We prove that p R, `q00 M `R ¨p R, `q00 M is the smallest M -type-definable two-sided ideal of R of bounded index, which we denote by R00M . Then S in the first sentence of the abstract is just R{ R00M and f : R Ñ R{ R00 M is the quotient map. In fact, f is the universal "definable" (in a suitable sense) locally compact model.