Remarks on theories of free algebras and modules
Abstract
We ask some questions and make some observations about the (complete) theory T (infinity, V) of free algebras in V on infinitely many generators, where V is a variety in the sense of universal algebra. We focus on the case T(infinity, R) where V is the variety of R-modules (R a ring). Building on work in Kucera-Pillay we characterize when all models of T(infinity, R) are free, projective, flat, as well as when T(infinity,R) is categorical in a higher power.
Cite
@article{arxiv.2004.10351,
title = {Remarks on theories of free algebras and modules},
author = {Anand Pillay and Philipp Rothmaler},
journal= {arXiv preprint arXiv:2004.10351},
year = {2020}
}
Comments
10 pages. In this new version we clarify the status of the conjecture of Tarski that we heard about from Baldwin. Actually the paper of Baldwin and Lachlan referred to in the paper already gives a counterexample to the conjecture, although we give another one which is possibly more natural: a variety of R-modules