Generalized Injectivity and Approximations - Sahinkaya, Serap; Trlifaj, Jan (Taylor & Francis Inc, 2016)
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 2; in this case, however, R may have an arbitrary representation type.
Keywords: Generalized injective modules; Module approximations; Pure-injective modules