Phd Degree / Doktora

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

Browse

Filters

2012 - 20193

Settings

Department: search.filters.department.Thesis (Doctoral)--İzmir Institute of Technology, MathematicsSubject: search.filters.subject.Modules (Algebra)

Search Results

Now showing 1 - 3 of 3
  • 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
    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.