Browsing by Author "Alizade, Rafail"

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Absolutely Supplement and Absolutely Complement Modules
(Izmir Institute of Technology, 2004) Erdoğan, Sultan Eylem; Alizade, Rafail; Alizade, Refail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We introduce and study absolutely supplement (respectively complement) modules. We call a module an absolutely supplement (respectively complement) if it is a supplement (respectively complement) in every module containing it. We show that a module is absolutely supplement (respectively complement) if and only if it is a supplement (respectively complement) in its injective envelope. The class of all absolutely supplement (respectively complement) modules is closed under extensions and under supplement submodules (respectively under factor modules by complement submodules). We also consider the dual notions of absolutely co-supplements (respectively co-complements).
Confinitely Amply Weakly Supplemented Modules
(Izmir Institute of Technology, 2005) Menemen, Filiz; Alizade, Rafail; Alizade, Rafail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We 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.
Proper Class Generated by Submodules That Have Supplements
(Izmir Institute of Technology, 2008) Demirci, Yılmaz Mehmet; Alizade, Rafail; Alizade, Rafail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
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.
Submodules That Have Supplements
(Izmir Institute of Technology, 2007) Çeliköz, Zafer; Alizade, Rafail; Alizade, Rafail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
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.