Mathematics / Matematik

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

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Start date: 2001Institution Author: search.filters.institutionAuthor.Büyükaşık, Engin

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Now showing 1 - 10 of 28
  • Article
    On Purities Relative To Minimal Right Ideals
    (Pleiades Publishing, 2023) Alagöz, Yusuf; Alizade, Rafail; Büyükaşık, Engin; Sağbaş, Selçuk
    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: 3
    Citation - Scopus: 3
    Dual Kasch rings
    (World Scientific Publishing, 2023) Lomp, Christian; Büyükaşık, Engin; Yurtsever, Haydar Baran
    It is well known that a ring R is right Kasch if each simple right R-module embeds in a projective right R-module. In this paper we study the dual notion and call a ring R right dual Kasch if each simple right R-module is a homomorphic image of an injective right R-module. We prove that R is right dual Kasch if and only if every finitely generated projective right R-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field k, a finite dimensional k-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.
  • Article
    Citation - WoS: 2
    Citation - Scopus: 2
    On simple-injective modules
    (World Scientific Publishing, 2022) Alagöz, Yusuf; Benli Göral, Sinem; Büyükaşık, Engin
    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: 5
    Citation - Scopus: 5
    On simple-direct modules
    (Taylor and Francis Ltd., 2021) Büyükaşık, Engin; Demir, Özlem; Diril, Müge
    Recently, in a series of papers “simple” versions of direct-injective and direct-projective modules have been investigated. These modules are termed as “simple-direct-injective” and “simple-direct-projective,” respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given. © 2020 Taylor & Francis Group, LLC.
  • Article
    On the Structure of Modules Defined by Opposites of Fp Injectivity
    (Springer Verlag, 2019) Büyükaşık, Engin; Kafkas Demirci, Gizem
    Let R be a ring with unity and let MR and RN be right and left modules,respectively. The module MR is said to be absolutely RN-pure if M circle times NL circle times N is amonomorphism for every extension LR of MR. For a module MR, the subpurity domain of MR is defined to be the collection of all modules RN, such that MR is absolutely RN-pure. Clearly, MR is absolutely RF-pure for every flat module RF and that MR is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, MR is said to be a test for flatness by subpurity (or t.f.b.s. for short) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. We characterize the structure of t.f.b.s. modules over commutative hereditary Noetherian rings. We prove that a module M is t.f.b.s. over a commutative hereditary Noetherian ring if and only if M/Z(M) is t.f.b.s. if and only if Hom(M/Z(M),S)0 for each singular simple module S. Prufer domains are characterized as those domains all of whose nonzero finitely generated ideals are t.f.b.s.
  • Article
    Citation - WoS: 8
    Citation - Scopus: 8
    On the Structure of Modules Defined by Subinjectivity
    (World Scientific Publishing, 2019) Altınay, Ferhat; Büyükaşık, Engin; Durgun, Yılmaz
    The aim of this paper is to present new results and generalize some results about indigent modules. The commutative rings whose simple modules are indigent or injective are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. We show that a reduced module is indigent if and only if it is a Whitehead test module for injectivity over commutative hereditary noetherian rings. Furthermore, Dedekind domains are characterized by test modules for injectivity by subinjectivity.
  • Article
    Rings and Modules Characterized by Opposites of Fp-Injectivity
    (Korean Mathematical Society, 2019) Büyükaşık, Engin; Kafkas Demirci, Gizem
    Let R be a ring with unity. Given modules M-R and N-R, M-R is said to be absolutely N-R-pure if M circle times N -> L circle times N is a monomorphism for every extension L-R of M-R. For a module M-R, the subpurity domain of M-R is defined to be the collection of all modules N-R such that M-R is absolutely N-R-pure. Clearly M-R is absolutely F-R-pure for every flat module F-R, and that M-R is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, M-R is said to be a test for flatness by subpurity (or t.f.b.s. for short) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. Every ring has a right t.f.b.s. module. R-R is t.f.b.s. and every finitely generated right ideal is finitely presented if and only if R is right semihereditary. A domain R is Priifer if and only if R is t.f.b.s. The rings whose simple right modules are t.f.b.s. or injective are completely characterized. Some necessary conditions for the rings whose right modules are t.f.b.s. or injective are obtained.
  • Article
    Citation - WoS: 8
    Citation - Scopus: 9
    Max-Projective Modules
    (World Scientific Publishing, 2020) Alagöz, Yusuf; Büyükaşık, Engin
    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.
  • Article
    Citation - WoS: 12
    Citation - Scopus: 11
    Poor Modules With No Proper Poor Direct Summands
    (Academic Press Inc., 2018) Alizade, Rafail; Büyükaşık, Engin; López-Permouth, Sergio; Yang, Liu
    As a mean to provide intrinsic characterizations of poor modules, the notion of a pauper module is introduced. A module is a pauper if it is poor and has no proper poor direct summand. We show that not all rings have pauper modules and explore conditions for their existence. In addition, we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity: one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule. The second condition holds for the ring of integers and is just as significant as the first one for Noetherian rings since, in that context, modules having poor pure submodules must themselves be poor. It is shown that the existence of paupers is equivalent to the Noetherian condition for rings with no middle class. As indecomposable poor modules are pauper, we study rings with no indecomposable right middle class (i.e. the ring whose indecomposable right modules are pauper or injective). We show that semiartinian V-rings satisfy this property and also that a commutative Noetherian ring R has no indecomposable middle class if and only if R is the direct product of finitely many fields and at most one ring of composition length 2. Structure theorems are also provided for rings without indecomposable middle class when the rings are Artinian serial or right Artinian. Rings for which not having an indecomposable middle class suffices not to have a middle class include commutative Noetherian and Artinian serial rings. The structure of poor modules is completely determined over commutative hereditary Noetherian rings. Pauper Abelian groups with torsion-free rank one are fully characterized.
  • Article
    Citation - WoS: 12
    Citation - Scopus: 11
    Rugged Modules: the Opposite of Flatness
    (Taylor and Francis Ltd., 2018) Büyükaşık, Engin; Enochs, Edgar; Rozas, J. R. García; Kafkas Demirci, Gizem; López-Permouth, Sergio; Oyonarte, Luis
    Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M+ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.