1 Ocak 2022 Dergi makalesi Açık Erişim

Aydogdu, Pinar; Durgun, Yilmaz

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  <identifier identifierType="URL">https://aperta.ulakbim.gov.tr/record/262163</identifier>
  <creators>
    <creator>
      <creatorName>Aydogdu, Pinar</creatorName>
      <givenName>Pinar</givenName>
      <familyName>Aydogdu</familyName>
      <affiliation>Hacettepe Univ, Dept Math, Ankara, Turkey</affiliation>
    </creator>
    <creator>
      <creatorName>Durgun, Yilmaz</creatorName>
      <givenName>Yilmaz</givenName>
      <familyName>Durgun</familyName>
      <affiliation>Cukurova Univ, Dept Math, Adana, Turkey</affiliation>
    </creator>
  </creators>
  <titles>
    <title>The Opposite Of Injectivity By Proper Classes</title>
  </titles>
  <publisher>Aperta</publisher>
  <publicationYear>2022</publicationYear>
  <dates>
    <date dateType="Issued">2022-01-01</date>
  </dates>
  <resourceType resourceTypeGeneral="Text">Journal article</resourceType>
  <alternateIdentifiers>
    <alternateIdentifier alternateIdentifierType="url">https://aperta.ulakbim.gov.tr/record/262163</alternateIdentifier>
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  <relatedIdentifiers>
    <relatedIdentifier relatedIdentifierType="DOI" relationType="IsIdenticalTo">10.2989/16073606.2022.2109221</relatedIdentifier>
  </relatedIdentifiers>
  <rightsList>
    <rights rightsURI="http://www.opendefinition.org/licenses/cc-by">Creative Commons Attribution</rights>
    <rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
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  <descriptions>
    <description descriptionType="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.</description>
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