Phd Degree / Doktora

Permanent URI for this collectionhttps://hdl.handle.net/11147/2869

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Now showing 1 - 7 of 7
  • Doctoral Thesis
    Belirli Altmodülleri Direkt Toplananlarına İzomorfik Olan Modüller
    (2025) Demir, Özlem Irmak; Büyükaşık, Engin
    Bir sağ R-modülüne, her sonlu olarak üretilen alt modülü direkt toplanan olduğunda kuvvetli düzenli denir. Bu tezin temel amacı, bazı kuvvetli düzenli modül sınıflarını eşitlik yerine izomorfizma kavramı temelinde ele alan farklı bir bakış açısıyla incelemek ve bu modüllerin yapısal özelliklerini araştırmaktır. Bu bağlamda, bir sağ R-modülü M, her (sonlu olarak üretilen) devirsel alt modülü M'nin bir direkt toplananı ile izomorfik ise (kuvvetli) sanal düzenli olarak adlandırılır. Ayrıca, M'nin tüm alt modülleri sanal düzenli ise M'ye tamamen sanal düzenli modül denir. Bu tezde yukarıda sözü edilen modüllerin keyfi bir halka üzerindeki temel özellikleri incelenmiş ve belirli bazı halkalar üzerindeki yapıları ve karakterizasyonları verilmiştir. Özellikle, değişmeli halkalar, tamlık bölgeleri, Dedekind bölgeleri ve değerlendirme bölgeleri üzerinde bu modüllerin yapıları incelenmiştir. Değerlendirme bölgeleri üzerinde, sonlu sunumlu (kuvvetli) sanal düzenli ve tamamen sanal düzenli modülleri yapıları tam olarak belirlenmiştir. Bu modüllerin benzer başka modüller ile ilişkileri de incelenmiştir.
  • Doctoral Thesis
    When Certain Relative Projectivity and Injectivity Conditions Imply the Global Projectivity and Injectivity
    (Izmir Institute of Technology, 2022) Benli Göral, Sinem; Büyükaşık, Engin
    A right R-module M is called R-projective provided that it is projective relative to the right R-module RR. One of the parts of this thesis deals with the rings whose all nonsingular right modules are R-projective. For a right nonsingular ring R, we prove that RR is of finite Goldie rank and all nonsingular right R-modules are R-projective if and only if R is right finitely Σ-CS and flat right R-modules are R-projective. Then, R-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that R-projectivity of nonsingular injective right modules is equivalent to R-projectivity of the injective hull E(RR). As a second goal, we deal with simple-injective modules. 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 quasi-Frobenius 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 quasi-Frobenius and B is hereditary. An abelian group is simpleinjective if and only if its torsion part is injective.
  • Doctoral Thesis
    On Relative Projectivity of Some Classes of Modules
    (Izmir Institute of Technology, 2019) Alagöz, Yusuf; Büyükaşık, Engin
    The main purpose of this thesis is to study R-projectivity and max-projectivity of some classes of modules, and module classes related to max-projective modules. 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 π:R → R/I. We call a ring R right almost-QF (resp. right max-QF) if every injective right R-module is R-projective (resp. max-projective). In this thesis we attempt to understand the class of right almost-QF (resp. right max-QF) rings. Among other results, we prove that a right Hereditary right Noetherian ring R is right almost-QF if and only if R is right max-QF if and only if R = S x T , where S is semisimple Artinian and T is right small. A right Hereditary ring is max-QF if and only if every injective simple right R-module is projective. Furthermore, a commutative Noetherian ring R is almost-QF if and only if R is max-QF if and only if R = A x B, where A is QF and B is a small ring. Moreover, we introduced and studied some homological objects related with max-projective modules.
  • Doctoral Thesis
    Modules Satisfying Conditions That Are Opposites of Absolute Purity and Flatness
    (Izmir Institute of Technology, 2017) Kafkas Demirci, Gizem; Büyükaşık, Engin
    The main purpose of this thesis is to study the properties which are opposites of absolute pure and flat modules. A right module M is said to be test for flatness by subpurity (for short, t.f.b.s.) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. A left module M is said to be rugged if its flatness domain is the class of all regular right R-modules. Every ring has a t.f.b.s. module. For a right Noetherian ring R every simple right R-module is t.f.b.s. or absolutely pure if and only if R is a right V-ring or R A×B, where A is right Artinian with a unique non-injective simple right R-module and Soc(AA) is homogeneous and B is semisimple. A characterization of t.f.b.s. modules over commutative hereditary Noetherian rings is given. Rings all (cyclic) modules of whose are rugged are shown to be von Neumann regular rings. 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 contains no properly nonzero ideals. Connections between rugged and poor modules are shown. Rugged Abelian groups are fully characterized.
  • Doctoral Thesis
    Homological Objects of Proper Classes Generated by Simple Modules
    (Izmir Institute of Technology, 2014) Durğun, Yılmaz; Büyükaşık, Engin
    The main purpose of this thesis is to study some classes of modules determined by neat, coneat and s-pure submodules. A right R-module M is called neat-flat (resp. coneat-flat) if the kernel of any epimorphism Y → M → 0 is neat (resp. coneat) in Y. A right R-module M is said to be absolutely s-pure if it is s-pure in every extension of it. If R is a commutative Noetherian ring, then R is C-ring if and only if coneat-flat modules are flat. A commutative ring R is perfect if and only if coneat-flat modules are projective. R is a right Σ -CS ring if and only if neat-flat right R-modules are projective. R is a right Kasch ring if and only if injective right R-modules are neat-flat if and only if the injective hull of every simple right R-module is neat-flat. If R is a right N-ring, then R is right Σ -CS ring if and only if pure-injective neat-flat right R-modules are projective if and only if absolutely s-pure left R-modules are injective and R is right perfect. A domain R is Dedekind if and only if absolutely s-pure modules are injective. It is proven that, for a commutative Noetherian ring R, (1) neat-flat modules are flat if and only if absolutely s-pure modules are absolutely pure if and only if R A × B, wherein A is QF-ring and B is hereditary; (2) neat-flat modules are absolutely s-pure if and only if absolutely s-pure modules are neat-flat if and only if R A × B, wherein A is QF-ring and B is Artinian with J2(B) = 0.
  • Doctoral Thesis
    Co-Coatomically Supplemented Modules
    (Izmir Institute of Technology, 2013) Güngör, Serpil; Büyükaşık, Engin; Büyükaşık, Engin
    The purpose of this study to define co-coatomically supplemented modules, -cocoatomically supplemented modules, co-coatomically weak supplemented modules and co-coatomically amply supplemented modules and examine them over arbitrary rings and over commutative Noetherian rings, in particular over Dedekind domains. Motivated by cofinite submodule which is defined by R. Alizade, G. Bilhan and P. F. Smith, we define co-coatomic submodule. A proper submodule is called co-coatomic if the factor module by this submodule is coatomic. Then we define co-coatomically supplemented module. A module is called co-coatomically supplemented if every co-coatomic submodule has a supplement in this module. Over a discrete valuation ring, a module is co-coatomically supplemented if and only if the basic submodule of this module is coatomic. Over a non-local Dedekind domain, if a reduced module is co-coatomically amply supplemented then the factor module of this module by its torsion part is divisible and P-primary components of this module are bounded for each maximal ideal P. Conversely, over a non-local Dedekind domain, if the factor module of a reduced module by its torsion part is divisible and P-primary components of this module are bounded for each maximal ideal P, then this module is co-coatomically supplemented. A ring R is left perfect if and only if any direct sum of copies of the ring is -co-coatomically supplemented left R-module. Over a discrete valuation ring, co-coatomically weak supplemented and co-coatomically supplemented modules coincide. Over a Dedekind domain, if the torsion part of a module has a weak supplement in this module, then the module is co-coatomically weak supplemented if and only if the torsion part is co-coatomically weak supplemented and the factor module of the module by its torsion part is co-coatomically weak supplemented. Every left R-module is co-coatomically weak supplemented if and only if the ring R is left perfect.
  • Doctoral Thesis
    Operations on Proper Classes Related To Supplements
    (Izmir Institute of Technology, 2012) Demirci, Yılmaz Mehmet; Büyükaşık, Engin
    The purpose of this study is to understand the properties of the operations +, ◦, and * defined on classes of short exact sequences and apply them to the proper classes related to supplements. The operation ◦ on classes of short exact sequences is introduced and it is proved that the class of extended weak supplements is the result of the operation ◦ applied to two classes one of which is the class of splitting short exact sequences. Using the direct sum of proper classes defined by R. Alizade, G. Bilhan and A. Pancar, a direct sum decomposition for quasi-splitting short exact sequences over the ring of integers is obtained. Closures of classes of short exact sequences along with the one studied by C. P. Walker, N. Hart and R. Alizade are defined over an integral domain. It is shown that these classes are proper when the underlying class is proper and they are related to the operation +. The closures of proper classes related to supplements are described in terms of Ivanov classes. Closures for modules over an integral domain are also defined and it is proved that submodules of torsion-free modules have unique closures. A closure for classes of short exact sequences is defined over an associative ring with identity and it is proved that this closure is proper when the underlying class is proper. Results shows that the operation + and closures of splitting short exact sequences plays an important role on the closures of proper classes.