WSA-Supplements and Proper Classes

Creative Commons License


MATHEMATICS, vol.10, no.16, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 10 Issue: 16
  • Publication Date: 2022
  • Doi Number: 10.3390/math10162964
  • Journal Name: MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Keywords: proper class of short exact sequences, wsa-supplement submodule, weakly semiartinian module, C-ring, CC-ring
  • Abdullah Gül University Affiliated: Yes


In this paper, we introduce the concept of wsa-supplements and investigate the objects of the class of short exact sequences determined by wsa-supplement submodules, where a submodule U of a module M is called a wsa-supplement in M if there is a submodule V of M with U + V = M and U boolean AND V is weakly semiartinian. We prove that a module M is weakly semiartinian if and only if every submodule of M is a wsa-supplement in M. We introduce CC-rings as a generalization of C-rings and show that a ring is a right CC-ring if and only if every singular right module has a crumbling submodule. The class of all short exact sequences determined by wsa-supplement submodules is shown to be a proper class which is both injectively and co-injectively generated. We investigate the homological objects of this proper class along with its relation to CC-rings.