Abstract
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 C1, C2, C3, quasi-continuous, continuous, and quasi-injective modules. We show that, except for the class of all C1-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 C1-modules over a right noetherian ring R is (pre)enveloping iff R is a certain right artinian ring of Loewy length ≤2; in this case, however, R may have an arbitrary representation type.
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