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arXiv · 1601.01101

Generalized injectivity and approximations

Serap Sahinkaya·Jan Trlifaj
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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.

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BibTeXRIS

Serap Sahinkaya, Jan Trlifaj. 2016-01-06. Generalized injectivity and approximations. https://doi.org/10.1080/00927872.2015.1087548

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