  We investigate the question of when free structures of infinite rank (in a
variety) possess model-theoretic properties like categoricity in higher power,
saturation, or universality. Concentrating on left $R$-modules we show, among
other things, that the free module of infinite rank $R^{(\kappa)}$ embeds every
$\kappa$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass
module corresponding to a descending chain of principal right ideals, we
construct a model of the theory $T$ of $R^{(\kappa)}$ whose projectivity is
equivalent to left perfectness, which allows to add a "stronger" equivalent
condition: $R^{(\kappa)}$ embeds every $\kappa$-generated flat left $R$-module
which is a model of $T$.
  In addition, we extend the model-theoretic construction of this Bass module
to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of
pure-projective modules. We put this new theory to use by reproving an old
result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.
