ABELIAN-GROUPS WITH MODULAR GENERIC

Let G be a stable abelian group with regular modular generic. We show that either 1. there is a definable nongeneric K less-than-or-equal-to G such that G/K has definable connected component and so strongly regular generics, or 2. distinct elements of the division ring yielding the dependence relation are represented by subgroups of G x G realizing distinct strong types (when regarded as elements of G(eq)). In the latter case one can choose almost 0-definable subgroups representing the elements of the division ring. We find a bound ((G : G(o))) 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 dependent of the particular model G as long as G/H is infinite.