  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.
