Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Proper Class Generated by Submodules That Have Supplements(Izmir Institute of Technology, 2008) Demirci, Yılmaz Mehmet; Alizade, Rafail; Alizade, RafailIn 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, RafailIn 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 Generalization of Cofinitely Supplemented Modules To Lattices(Izmir Institute of Technology, 2005) Çetindil, Yasin; Alizade, RafailIn 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 modulesMaster Thesis Confinitely Amply Weakly Supplemented Modules(Izmir Institute of Technology, 2005) Menemen, Filiz; Alizade, Rafail; Alizade, RafailWe study amply weak supplemented modules and co¯nitely amply weakly supple-mented modules in this thesis. We prove that every factor module, homomorphic image, supplemented submodule of an amply (co¯nitely) weak supplemented module is amply (co¯nitely) weak supplemented.Master Thesis The Least Proper Class Containing Weak Supplement(Izmir Institute of Technology, 2009) Durğun, Yılmaz; Alizade, RafailThe main purpose of this thesis is to investigate the least proper class containing the classWS of R-modules determined by weak supplement submodules over a ring R, in particular, over hereditary rings. A submodule A of a module B has(is) weak supplement if and only if there exist a submodule V in B such that A + V . B and the intersection of submodules of A and V is small in B. The classWS does not form a proper class, in general. By extending the class WS, we obtained the least proper class containing the class WS of R-modules over hereditary rings. We investigate the homological objects of the least proper class. We determine the structure of elements of the proper class by submodules.