| Résume | I will present several fundamental results about generating ideals in finitely many steps inside additive groups of rings from my joint paper with T. Rzepecki. I will also mention an application to computations of definable and classical Bohr compactifications of the groups of upper unitriangular and invertible upper triangular matrices over arbitrary unital rings, based on my joint paper with J. Gismatullin and G. Jagiella. An essential role in this research is played by model-theoretic connected components of definable groups and rings. In particular, these components are used to compute the above Bohr compactifications. Regarding connected components, roughly speaking, one of our main results says that the type-definable connected component of the additive subgroup of a definable (saturated) unital ring generates an ideal in finitely many steps (and so this generated ideal is exactly the ring type-definable connected component). |
