Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection

Permanent URI for this collectionhttps://hdl.handle.net/11147/7148

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Has files: YesPublisher: search.filters.publisher.World Scientific Publ Co Pte Ltd

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  • Article
    Projectivity and Quasi-Projectivity With Respect To Epimorphisms To Simple Modules
    (World Scientific Publ Co Pte Ltd, 2025) Alagoz, Yusuf; Alizade, Rafail; Buyukasik, Engin
    Using the notion of relative max-projectivity, max-projectivity domain of a module is investigated. Such a domain includes the class of all modules whose maximal submodules are direct summands (this class denoted as MDMod -R). We call a module max-p-poor if its max-projectivity domain is exactly the class MDMod -R. We establish the existence of max-p-poor modules over any ring. Furthermore, we study commutative rings whose simple modules are projective or max-p-poor. Additionally, we determine the right Noetherian rings for which all right modules are projective or p-poor. Max-p-poor abelian groups are fully characterized and shown to coincide precisely with p-poor abelian groups. We also further investigate modules that are max-projective relative to themselves, which are known as simple-quasi-projective modules. Several properties of these modules are provided, and the structure of certain classes of simple-quasi-projective modules is determined over specific commutative rings including the ring of integers and valuation domains.
  • Article
    Virtually Regular Modules
    (World Scientific Publ Co Pte Ltd, 2025) Buyukasik, Engin; Demir, Ozlem Irmak
    In this paper, we call a right module M (strongly) virtually regular if every (finitely generated) cyclic submodule of M is isomorphic to a direct summand of M. M is said to be completely virtually regular if every submodule of M is virtually regular. In this paper, characterizations and some closure properties of the aforementioned modules are given. Several structure results are obtained over commutative rings. In particular, the structures of finitely presented (strongly) virtually regular modules and completely virtually regular modules are fully determined over valuation domains. Namely, for a valuation domain R with the unique nonzero maximal ideal P, we show that finitely presented (strongly) virtually regular modules are free if and only if P is not principal; and that P = Rp is principal if and only if finitely presented virtually regular modules are of the form R-n circle plus (R/Rp)(n)(1) circle plus (R/Rp(2))(n)(2) circle plus center dot center dot center dot circle plus (R/Rp(k))(n)(k) for nonnegative integers n, k, n(1), n(2),...,n(k). Similarly, we prove that P = Rp is principal if and only if finitely presented strongly virtually regular modules are of the form R-n circle plus (R/Rp)(m), where m,n are nonnegative integers. We also obtain that, R admits a nonzero finitely presented completely virtually regular module M if and only if P = Rp is principal. Moreover, for a finitely presented R-module M, we prove that: (i) if R is not a DVR, then M is completely virtually regular if and only if M congruent to( R/Rp)(m); and (ii) if R is a DVR, then M is completely virtually regular if and only if M congruent to R-n circle plus ( R/Rp)(m). Finally, we obtain a characterization of finitely generated virtually regular modules over the ring of integers.