Generalized injectivity and approximations
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2016
- DOI:
- 10.48550/arXiv.1601.01101
- arXiv:
- arXiv:1601.01101
- Bibcode:
- 2016arXiv160101101S
- Keywords:
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- Mathematics - Representation Theory;
- Primary: 16D50. Secondary: 18G25;
- 16D70
- E-Print:
- Commun. Algebra 44(2016), 4047-4055