  Injective, pure-injective and fp-injective modules are well known to provide
for approximations in the category Mod-R for an arbitrary ring R. We prove that
this fails for many other generalizations of injectivity: the $C_1$, $C_2$,
$C_3$, quasi-continuous, continuous, and quasi-injective modules. We show that,
except for the class of all $C_1$-modules, each of the latter classes provides
for approximations only when it coincides with the injectives (for
quasi-injective modules, this forces R to be a right noetherian V-ring, in the
other cases, R even has to be semisimple artinian). The class of all
$C_1$-modules over a right noetherian ring R is (pre)enveloping, iff R is a
certain right artinian ring of Loewy length at most 2; in this case, however, R
may have an arbitrary representation type.

Representation Theory