Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Author: search.filters.author.Alagöz, YusufInstitution Author: search.filters.institutionAuthor.Büyükaşık, Engin

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  • Article
    On Purities Relative To Minimal Right Ideals
    (Pleiades Publishing, 2023) Alagöz, Yusuf; Büyükaşık, Engin; Alizade, Rafail; Alizade, Rafail; Büyükaşık, Engin; Alagöz, Yusuf; Sağbaş, Selçuk; 04.02. Department of Mathematics; 01. Izmir Institute of Technology; 04. Faculty of Science
    Abstract: We call a right module M weakly neat-flat if (Formula presented.) is surjective for any epimorphism (Formula presented.) and any simple right ideal S . A left module M is called weakly absolutely s-pure if (Formula presented.) is monic, for any monomorphism (Formula presented.) and any simple right ideal S . These notions are proper generalization of the neat-flat and the absolutely s-pure modules which are defined in the same way by considering all simple right modules of the ring, respectively. In this paper, we study some closure properties of weakly neat-flat and weakly absolutely s-pure modules, and investigate several classes of rings that are characterized via these modules. The relation between these modules and some well-known homological objects such as projective, flat, injective and absolutely pure are studied. For instance, it is proved that R is a right Kasch ring if and only if every weakly neat-flat right R -module is neat-flat (moreover if R is right min-coherent) if and only if every weakly absolutely s-pure left R -module is absolutely s-pure. The rings over which every weakly neat-flat (resp. weakly absolutely s-pure) module is injective and projective are exactly the QF rings. Finally, we study enveloping and covering properties of weakly neat-flat and weakly absolutely s-pure modules. The rings over which every simple right ideal has an epic projective envelope are characterized. © 2023, Pleiades Publishing, Ltd.
  • Article
    Citation - WoS: 2
    Citation - Scopus: 2
    On simple-injective modules
    (World Scientific Publishing, 2022) Alagöz, Yusuf; Benli Göral, Sinem; Büyükaşık, Engin; Alagöz, Yusuf; 01. Izmir Institute of Technology; 04.02. Department of Mathematics; 04. Faculty of Science
    For a right module M, we prove that M is simple-injective if and only if M is min-N-injective for every cyclic right module N. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is QF if and only if simple-injective right modules are projective. For a commutative Noetherian ring R, we prove that every finitely generated simple-injective R-module is projective if and only if R = A × B, where A is QF and B is hereditary. An abelian group is simple-injective if and only if its torsion part is injective. We show that the notions of simple-injective, strongly simple-injective, soc-injective and strongly soc-injective coincide over the ring of integers.
  • Article
    Citation - WoS: 8
    Citation - Scopus: 9
    Max-Projective Modules
    (World Scientific Publishing, 2020) Alagöz, Yusuf; Alagöz, Yusuf; Büyükaşık, Engin; Büyükaşık, Engin; 01. Izmir Institute of Technology; 04.02. Department of Mathematics; 04. Faculty of Science
    Weakening the notion of R-projectivity, a right R-module M is called max-projective provided that each homomorphism f: M ? R/I, where I is any maximal right ideal, factors through the canonical projection : R ? R/I. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are R-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring R, we prove that injective modules are R-projective if and only if R = A × B, where A is QF and B is a small ring. If R is right hereditary and right Noetherian then, injective right modules are max-projective if and only if R = S × T, where S is a semisimple Artinian and T is a right small ring. If R is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective. © 2020 World Scientific Publishing Company.