The opposite of injectivity by proper classes

Abstract

Proper classes (or exact structures) offer rich research topics due to their important role in category theory. Motivated by the studies on  opposite of injective modules, we introduce a new approach to opposed to injectivity in terms of injectively generated proper classes. The  smallest possible proper class generated injectively by a single module is the class of all split short exact sequences. We call a  module M ι-indigent if the proper class injectively generated by M consists only of split short exact sequences. We are able to show that if  R is a ring which is not von Neumann regular, then every right (pure-injective) R-module is either injective or ι-indigent if and only if R is  an Artinian serial ring with J ² (R) = 0 and has a unique non-injective simple right R-module up to isomorphism. Moreover, if R is a ring  such that every simple right R-module is pure-injective, then every simple right R-module is ι-indigent or injective if and only if R is either  a right V -ring or R = A × B, where A is semisimple, and B is an Artinian serial ring with J ² (B) = 0. We investigate the class ι(R) which  consists of those proper classes P such that P is injectively generated by a module. We call such a class (right) proper injective profile of a  ring R. We prove that if R is an Artinian serial ring with J ² (R) = 0, then |ι(R)| = 2ⁿ , where n is the number of non-isomorphic non-injective  simple right R-modules. In addition, if ι(R) is a chain, then R is a right Noetherian ring over which every right R-module is either  projective or i-test, and has a unique singular simple right R-module. Furthermore, in this case, R is either right hereditary or right Kasch.  We observe that |ι(R)| ̸= 3 for any ring R which is not von Neumann regular. We construct a bounded complete lattice structure on ι(R) in  case ι(R) is a partially ordered set under set inclusion. Moreover, if R is an Artinian serial ring with J ² (R) = 0, then this lattice structure is  Boolean.