Master Degree / Yüksek Lisans Tezleri

Permanent URI for this collectionhttps://hdl.handle.net/11147/3008

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

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Now showing 1 - 10 of 12
  • Master Thesis
    Characterizations of Simple-Direct Modules
    (01. Izmir Institute of Technology, 2020) Diril, Müge; Büyükaşık, Engin; Durğun, Yılmaz
    In 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, Engin
    The 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
    Krull-Schmidt Properties Over Rings of Finite Character
    (Izmir Institute of Technology, 2016) Gürbüz, Ezgi; Ay Saylam, Başak
    The main purpose of this thesis is to investigate the notion of Krull-Schmidt properties over rings of finite character. In accordance with this aim, we give a survey of necessary and sufficient conditions on an h-local domain for certain Krull-Schmidt properties hold for direct sums of ideals, direct sums of indecomposable submodules of finitely generated free modules and direct sums of rank one torsion-free modules. By using obtained characterizations, some useful results for Krull-Schmidt properties of modules over Noetherian and Prüfer domains are proven. Besides, the characterizations of Noetherian UDI domains are given.
  • Master Thesis
    Strongly Noncosingular Modules
    (Izmir Institute of Technology, 2014) Alagöz, Yusuf; Büyükaşık, Engin
    The 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
    Modules Whith Coprimary Decomposition
    (Izmir Institute of Technology, 2009) Tekin, Semra; Pusat, Dilek
    This thesis presents the theory of coprimary decomposition of modules over a commutative noetherian ring and its coassociated prime ideals. This theory is first introduced in 1973 by I. G. Macdonald as a dual notion of an important tool of associated primes and primary decomposition in commutative algebra. In this thesis, we studied the basic properties of coassociated prime ideals to a module M and gathered some modules in the literature which have coprimary decomposition. For example, we showed that artinian modules over commutative rings are representable. Moreover if R is a commutative noetherian ring, then we showed that injective modules over R are representable. Finally, we discussed the uniqueness properties of coprimary decomposition.
  • Master Thesis
    Proper Class Generated by Submodules That Have Supplements
    (Izmir Institute of Technology, 2008) Demirci, Yılmaz Mehmet; Alizade, Rafail; Alizade, Rafail
    In this thesis, we study the class S of all short exact sequences 0 A B C 0 where Im& has a supplement in B, i.e. a minimal elemenr in the set {V B V + Im& . B}.The corresponding elements of ExtR(C;A) are called k-elements. In general k-elements need not form a subgroup in ExtR(C;A), but in the category TR of torsion R-modules over a Dedekind domain R, S is a proper class; there are no nonzero S-projective modules and the only S-injective modules are injective R-modules in TR. In this thesis we also give the structure of S-coinjective R-modules in TR. Moreover, we define the class SB of all short exact sequences 0 A B C 0 where Im & has a supplement V in B and V in B and In & is bounded. The corresponding elements of ExtR(C;A) are called B-elements. Over a noetherian integral domain of Krull dimension 1, B-elements form a proper class. In the category TR over a Dedekind domain R, SB is a proper class; there are no nonzero SB-projective R-modules and SB-injective R-modules are only the injective R-modules. In the category TR, reduced SB-coinjective R-modules are bounded R-modules.
  • Master Thesis
    Submodules That Have Supplements
    (Izmir Institute of Technology, 2007) Çeliköz, Zafer; Alizade, Rafail; Alizade, Rafail
    In this thesis we study theK -elements of extension modules where R is a principal ideal domain. In general K -elements need not form a submodule in an extension module but if C is divisible and almost all primary components of C are zero, they coincide with torsion elements of extension module. If C is divisible and torsion, not all primary components of C are zero, andAis torsion free ok rank 1 then a nonzero element of extension module is a K-element if and only if the type of the element in extension module is less than or equal to the type of A. Also we define B-elements which form a submodule of extension module and study their relation with K-elements.
  • Master Thesis
    Totally Weak Supplemented Modules
    (Izmir Institute of Technology, 2007) Top, Serpil; Alizade, Rafael
    The main purpose of this thesis is to give a survey about some classes of modules including supplemented, weakly supplemented, totally supplemented and totally weak supplemented modules over commutative Noetherian rings, in particular over Dedekind domains based on results of H. Zöschinger, P. Rudlof and P. F. Smith. A module is weakly supplemented if and only if the factor of that module by a finite direct sum of its hollow submodules is weakly supplemented. A module is weakly supplemented (totally weak supplemented) if and only if the factor of it by a linearly compact submodule is weakly supplemented (totally weak supplemented).
  • Master Thesis
    Strongly T-Noncosingular Modules
    (Izmir Institute of Technology, 2010) Günyüz, Ozan; Büyükaşık, Engin
    This 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.
  • Master Thesis
    Generalization of Cofinitely Supplemented Modules To Lattices
    (Izmir Institute of Technology, 2005) Çetindil, Yasin; Alizade, Rafail
    In this thesis we study how to extend the notion of co¯nitely supplemented module to lattice theory. A submodule N of a module M is called co¯nite if the factor module M/N is ¯nitely generated and we say that M is a co¯nitely supplemented module if every co¯nite submodule of M has a supplement. We analogously de¯ne the notions of co¯nite element and co¯nitely supplemented lattice for lattices. Inspired by the similarities between the properties of modules and modular lattices, we obtain results for co¯nitely supplemented modular lattices, analogous to results for co¯nitely supplemented modules