Department of Mathematics

This will be a very informal treatment of old material. I spent a lot of time thinking about this recently in relation to an old research problem, and although it probably will not be of help there, I thought that it might be worthwhile to talk about it. If time allows, I will also describe the old problem.

Exercise Let \( A \) and \( B \) be matrices over a field \( \mathbb{F} \); and let \( \overline{v} \), \( \overline{x} \) and \( \overline{x} \) vectors of variables; all of the appropriate shapes to make the following meaningful:
Under what natural conditions (beginning linear algebra) on the matrices does:

1. \( A\overline{v}=0 \) imply \( B\overline{v} =0 \)?
2. \( \exists\overline{x}\, B\overline{x}= \overline{v} \) imply \( \exists\overline{y}\, A\overline{y}= \overline{v} \)?

The parallel questions for modules over a ring \( R\) are much more complicated. Even over abelian groups there are difficulties. A positive primitive formula (ppf) is one of the form \( \varphi(\overline{v})\equiv \exists\overline{x}\, A\overline{x}= B\overline{v} \) where \( A \) and \( B \) are matrices over \( R\). Such formulas suffice to describe first order theories of modules. Let \( \varphi'(\overline{v}) \) be another such pp-formula.

Question: Under what conditions on the matrices \( A \), \( B \), \( A' \), and \( B' \) does does \( \varphi(\overline{v}) \) imply \( \varphi'(\overline{v})\)?

The answer has been well-known for some time, but is complicated; I will describe some of the reasons why the question is complicated, and describe the results due to Mike Prest (unpublished manuscript, 1983, but given in full in [1]).

References:

1. Prest, Mike, Model Theory and Modules, LMS Lecture Note Series 130 (1988), Section 8.3.