Generating ideals by additive subgroups 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+R⋅H contains an A-definable, two-sided ideal of finite index. As a corollary, we obtain a positive answer to Question 4.9 of [4]: if R is unital, then (R¯,+)A00+R¯⋅(R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00, which also implies that R¯A00=R¯A000, where R¯≻R is a sufficiently saturated elementary extension of R, (R¯,+)A00 [resp. R¯A00] denotes the smallest A-type-definable, bounded index additive subgroup [resp. ideal] of R¯, and R¯A000 is the smallest invariant over A, bounded index ideal of R¯. If R is of positive characteristic (not necessarily unital), we get a sharper result: (R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00. We obtain a similar result (but with more steps required) for finitely generated (not necessarily unital) rings. We obtain analogous results for topological rings. The above result for unital rings implies that the simplified descriptions of the definable (and so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 4.5 of [4] are valid for all unital rings.We also analyze many concrete examples, where we compute the number of steps needed to generate a group by (R¯∪{1})⋅(R¯,+)A00 and study related aspects, showing “optimality” of some of our main results and yielding answers to some natural questions.