Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Characterizations of Simple-Direct Modules(01. Izmir Institute of Technology, 2020) Diril, Müge; Büyükaşık, Engin; Durğun, YılmazIn this thesis, we study on simple-direct-injective and simple-direct-projective modules. We give a complete characterization of the aforementioned modules simple-direct-injective and simple-direct-projective modules over the ring of integers. 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. In addition, Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.Master Thesis On Generalization of Hopfian Modules(Izmir Institute of Technology, 2018) Yaman, Mehmet; Büyükaşık, EnginThe notion of Hopfian modules are defined as a generalization of modules of finite length as the modules whose surjective endomorphisms are isomorphisms. These modules and several generalizations of them are extensively studied in the literature. The aim of this thesis is to review some known results and extends some results about generalized Hopfian and weakly Hopfian modules. It is shown that a module is Hopfian if and only if it is both generalized Hopfian and weakly Hopfian. Torsion-free abelian groups are weakly Hopfian. Any nonsingular uniform module is weakly Hopfian. Direct summands of weakly Hopfian modules is weakly Hopfian. It is shown that direct sum weak Hopfian modules is not necessarily weakly Hopfian.Master Thesis Strongly Noncosingular Modules(Izmir Institute of Technology, 2014) Alagöz, Yusuf; Büyükaşık, EnginThe main purpose of this thesis is to investigate the notion of strongly noncosingular modules. We call a right R-module M strongly noncosingular if for every nonzero right R module N and every nonzero homomorphismf : M → N, Im(f) is not a cosingular (or Radsmall) submodule of N in the sense of Harada. It is proven that (1) A right R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary serial if and only if the class of injective right R-modules coincides with the class of (strongly) noncosingular right R-modules; (3) a right hereditary ring R is Max-ring if and only if absolutely coneat right R-modules are strongly noncosingular; (4) a commutative ring R is semisimple if and only if the class of injective R-modules coincides with the class of strongly noncosingular R-modules.Master Thesis Strongly T-Noncosingular Modules(Izmir Institute of Technology, 2010) Günyüz, Ozan; Büyükaşık, EnginThis thesis is mainly concerned with the T-noncosingularity issue of a module. Derya Keskin Tutuncu and Rachid Tribak introduced the T-noncosingular modules and gave some properties of these modules. A moduleM is said to be T-noncosingular relative to N if, for every nonzero homomorphism f from M to N, the image of f is not small in N. Inspired by this study, we define a new kind of module, as a particular case of T-noncosingular modules, and call it strongly T-noncosingular modules. We define M to be strongly T-noncosingular relative to N if, for every nonzero homomorphism f from M to N, the image of f is not contained in the radical of N. Obviously, if a module is strongly T-noncosingular, then it is also T-noncosingular, but the converse is, in general, not true. In an attempt to identify the situation when a T-noncosingular module is strongly T-noncosingular, we give necessary and sufficient conditions in terms of the specific ring structures as well as well-known module types.