THE OPPOSITE OF INJECTIVITY BY PROPER CLASSES


Aydogdu P., Durgun Y.

QUAESTIONES MATHEMATICAE, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2022
  • Doi Number: 10.2989/16073606.2022.2109221
  • Journal Name: QUAESTIONES MATHEMATICAE
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Keywords: Proper injective profile, injective modules, indigent modules, injectively generated proper classes, ALTERNATIVE PERSPECTIVE, MODULES, RINGS, PROJECTIVITY
  • Hacettepe University Affiliated: Yes

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 iota-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 iota-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 t-indigent or injective if and only if R is either a right V-ring or R = A x B, where A is semisimple, and B is an Artinian serial ring with J(2) (B) = 0. We investigate the class iota(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 vertical bar iota(R)vertical bar = 2(n), where n is the number of non-isomorphic non-injective simple right R-modules. In addition, if iota(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 vertical bar iota(R)vertical bar not equal 3 for any ring R which is not von Neumann regular. We construct a bounded complete lattice structure on iota(R) in case iota(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.