1 Ocak 2022 Dergi makalesi Açık Erişim

Aydogdu, Pinar; Durgun, Yilmaz

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{
  "@context": "https://schema.org/", 
  "@id": 262163, 
  "@type": "ScholarlyArticle", 
  "creator": [
    {
      "@type": "Person", 
      "affiliation": "Hacettepe Univ, Dept Math, Ankara, Turkey", 
      "name": "Aydogdu, Pinar"
    }, 
    {
      "@type": "Person", 
      "affiliation": "Cukurova Univ, Dept Math, Adana, Turkey", 
      "name": "Durgun, Yilmaz"
    }
  ], 
  "datePublished": "2022-01-01", 
  "description": "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.", 
  "headline": "THE OPPOSITE OF INJECTIVITY BY PROPER CLASSES", 
  "identifier": 262163, 
  "image": "https://aperta.ulakbim.gov.tr/static/img/logo/aperta_logo_with_icon.svg", 
  "license": "http://www.opendefinition.org/licenses/cc-by", 
  "name": "THE OPPOSITE OF INJECTIVITY BY PROPER CLASSES", 
  "url": "https://aperta.ulakbim.gov.tr/record/262163"
}