Abelian groups with modular generic

Journal of Symbolic Logic 56 (1):250-259 (1991)
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Let G be a stable abelian group with regular modular generic. We show that either 1. there is a definable nongeneric K ≤ 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 × 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 $((G: G^0))$ 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



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References found in this work

Unidimensional theories are superstable.Ehud Hrushovski - 1990 - Annals of Pure and Applied Logic 50 (2):117-137.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.
Locally finite weakly minimal theories.James Loveys - 1991 - Annals of Pure and Applied Logic 55 (2):153-203.
Classifiable theories without finitary invariants.E. Bouscaren & E. Hrushovski - 2006 - Annals of Pure and Applied Logic 142 (1-3):296-320.

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