Generating ideals by additive subgroups of rings
Krzysztof Krupiński, Tomasz Rzepecki
TL;DR
This work establishes a fundamental link between model-theoretic connected components of rings and their additive subgroups: for any A-definable finite-index subgroup H of (R,+), the set H+RH contains an A-definable finite-index two-sided ideal. This yields strong, uniform descriptions of definable Bohr compactifications for many rings, proving, in particular, that bar{R}^{0}_A = bar{R}^{00}_A = bar{R}^{000}_A for unital rings and rings of positive characteristic, with sharper 2-step results in the latter case. The authors develop a framework of equivalent finite-step-generation conditions, prove the main algebraic and topological theorems, and illustrate sharpness via numerous examples (including Z[X], exotic characteristic-2 rings, and Z2^ω) that delineate the boundaries of these phenomena. They also extend the results to finitely generated and topological rings, obtaining practical implications for Bohr compactifications of matrix groups over rings. Overall, the paper provides both deep structural insights and concrete, checkable criteria for when additive and ring-theoretic data control model-theoretic and topological compactifications in broad classes of rings.
Abstract
We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings. Let $R$ be any ring equipped with an arbitrary additional first order structure, and $A$ a set of parameters. We show that whenever $H$ is an $A$-definable, finite index subgroup of $(R,+)$, then $H+RH$ contains an $A$-definable, two-sided ideal of finite index. As a corollary, we positively answer Question 3.9 of [Bohr compactifications of groups and rings, J. Gismatullin, G. Jagiella and K. Krupiński]: if $R$ is unital, then $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$, where $\bar R \succ R$ is a sufficiently saturated elementary extension of $R$, and $(\bar R,+)^{00}_A$ [resp. $\bar R^{00}_A$] is the smallest $A$-type-definable, bounded index additive subgroup [resp. ideal] of $\bar R$. This implies that $\bar R^{00}_A=\bar R^{000}_A$, where $\bar R^{000}_A$ is the smallest invariant over $A$, bounded index ideal of $\bar R$. If $R$ is of finite characteristic (not necessarily unital), we get a sharper result: $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$. We obtain similar results for finitely generated (not necessarily unital) rings and for topological rings. The above results imply that the simplified descriptions of the definable (so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 3.5 of the aforementioned paper are valid for all unital rings. We analyze many examples, where we compute the number of steps needed to generate a group by $(\bar R \cup \{1\}) \cdot (\bar R,+)^{00}_A$ and study related aspects, showing "optimality" of some of our main results and answering some natural questions.