Generalized injectivity and approximations

TL;DR

This paper investigates various generalizations of injective modules and demonstrates that most do not provide for approximations unless they coincide with classical injectives, with specific conditions on the ring R.

Contribution

It shows that many generalized injective modules fail to provide approximations unless they are classical injectives, identifying precise conditions on the ring R.

Findings

Most generalized injective classes do not provide approximations.

Only C_1-modules over certain rings are (pre)enveloping.

R is semisimple artinian or a specific artinian ring for approximation properties.

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 $C_1$, $C_2$, $C_3$, 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.