Abelian groups with modular generic

Abstract

AbstractLet G be a stable abelian group with regular modular generic. We show that either1. there is a definable nongeneric K ≤ G such that G/K has definable connected component and so strongly regular generics, or2. distinct elements of the division ring yielding the dependence relation are represented by subgroups of G × G realizing distinct strong types (when regarded as elements of Geq).In the latter case one can choose almost 0-definable subgroups representing the elements of the division ring. We find a bound ((G: G0)) for the size of the division ring in case G has no definable subgroup K so that G/K is infinite with definable connected component. We show in case (2) that the group G/H, where H consists of all nongeneric points of G, inherits a weakly minimal group structure from G naturally, and Th(G/H) is independent of the particular model G as long as G/H is infinite.