Department of Mathematics

A submodule \( M \) of an \( R \)-module \( N \) is union-definable if it is the union of subgroups of \( N \) definable in the first-order theory of \( N \) as an \( R \)-module. If \( M \) is in fact itself definable, then the model theory of \( M \) and the model theory of \( N/M \) can be no more complicated than that of \( N \) itself.

The situation is more complicated when \( M \) is only definable as a proper union of subgroups. I am in particular interested in the structure and properties of \( N/M \), especially when \( N \) is a direct-sum indecomposable pure-injective module.

I'll start with an overview/review of the basic facts of the model theory of modules and a few examples, then work through some aspects of the following questions:

1. Given a definable subgroup \( A \) of \( N/M \), what is its inverse image in \( N \)?
2. Given definable subgroups \( A_1 > A_2 \) of \( N/M \), how do we describe the gap between their inverse images in \( N \)?
3. Given a descending sequence of definable subgroups of \( N/M \), how is this reflected in \( N \)?

All this is in aid of a hard problem that I've been working on for a long time now; I'll finish with some of the motivation and background of that problem.