Generating ideals by additive subgroups of rings

TL;DR

This paper explores how finite index ideals and additive subgroups in rings can be generated within finitely many steps, providing new insights into model-theoretic connected components and their applications to Bohr compactifications.

Contribution

It establishes fundamental results linking finite index subgroups and ideals in rings with model-theoretic connected components, extending to non-unital and topological rings, and computes generation steps in various examples.

Findings

Finite index subgroups plus ring multiplication generate finite index ideals.

Unital rings satisfy $(ar R,+)^{00}_A + ar R imes (ar R,+)^{00}_A + ar R imes (ar R,+)^{00}_A = ar R^{00}_A$.

Results apply to Bohr compactifications and show optimality in examples.

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. Krupi\'nski]: 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.