Let R be a ring and M any
(right) R-module. For any set I let M(I) and MI denote the direct sum and
respectively the direct product of copies of M indexed by the set I. For any cardinal
number r, let 𝒞r denote the class of R-modules admitting a generating set of
cardinality ≦ r. In this paper we study the relationship between the pure-exactness of
the sequence 0 → M(I)→ MI→ MI∕M(I)→ 0 with respect to 𝒞r under the functor
HomR and chain conditions on suitably defined families of R-modules. This study led
us to the introduction of five properties Ar, A(r), Dr, D(r), and Pr for any
R-module M. We also study the effect of base extension (both covariant and
contravariant) of the ring R on modules having any (or some) of the above
mentioned properties. Finally we obtain necessary and sufficient conditions
for
to be pure-exact with respect to 𝒞k, where {Mα} is any family of R-modules and k
any integer ≧ 1.