Alizade, Rafail
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Alizade, Refail
Alizade, R.
Alizade, R
Alizade, R.
Alizade, R
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04.02. Department of Mathematics
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Former Staff
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Documents
31
Citations
212
h-index
9
Documents
24
Citations
216
Scholarly Output
22
Articles
14
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15974/8846
Supervised MSc Theses
7
Supervised PhD Theses
0
WoS Citation Count
129
Scopus Citation Count
126
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0
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1
WoS Citations per Publication
5.86
Scopus Citations per Publication
5.73
Open Access Source
19
Supervised Theses
7
| Journal | Count |
|---|---|
| Communications in Algebra | 3 |
| Journal of Algebra | 2 |
| Hacettepe Journal of Mathematics and Statistics | 1 |
| Indian Journal of Pure and Applied Mathematics | 1 |
| Journal of Algebra and Its Applications | 1 |
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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 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.Article Projectivity and Quasi-Projectivity With Respect To Epimorphisms To Simple Modules(World Scientific Publ Co Pte Ltd, 2025) Alagoz, Yusuf; Alizade, Rafail; Buyukasik, EnginUsing the notion of relative max-projectivity, max-projectivity domain of a module is investigated. Such a domain includes the class of all modules whose maximal submodules are direct summands (this class denoted as MDMod -R). We call a module max-p-poor if its max-projectivity domain is exactly the class MDMod -R. We establish the existence of max-p-poor modules over any ring. Furthermore, we study commutative rings whose simple modules are projective or max-p-poor. Additionally, we determine the right Noetherian rings for which all right modules are projective or p-poor. Max-p-poor abelian groups are fully characterized and shown to coincide precisely with p-poor abelian groups. We also further investigate modules that are max-projective relative to themselves, which are known as simple-quasi-projective modules. Several properties of these modules are provided, and the structure of certain classes of simple-quasi-projective modules is determined over specific commutative rings including the ring of integers and valuation domains.Article Citation - WoS: 13Citation - Scopus: 14Poor and Pi-Poor Abelian Groups(Taylor and Francis Ltd., 2017) Alizade, Rafail; Büyükaşık, EnginIn this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to (Formula presented.) , where P is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely, it is proved that the direct sum of U(ℕ), where U ranges over all nonisomorphic uniform abelian groups, is pi-poor. Moreover, for a pi-poor abelian group M, it is shown that M can not be torsion, and each p-primary component of M is unbounded. Finally, we show that there are pi-poor groups which are not poor, and vise versa.Master Thesis Absolutely Supplement and Absolutely Complement Modules(Izmir Institute of Technology, 2004) Erdoğan, Sultan Eylem; Alizade, Rafail; Alizade, RefailWe 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).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.Article Citation - WoS: 10Citation - Scopus: 10Special Precovers in Cotorsion Theories(Cambridge University Press, 2002) Akıncı, Karen D.; Alizade, RafailA cotorsion theory is defined as a pair of classes Ext-orthogonal to each other. We give a hereditary condition (HC) which is satisfied by the (flat, cotorsion) cotorsion theory and give properties satisfied by arbitrary cotorsion theories with an HC. Given a cotorsion theory with an HC, we consider the class of all modules having a special precover with respect to the first class in the cotorsion theory and show that this class is closed under extensions. We then raise the question of whether this class is resolving or coresolving.Article Citation - WoS: 23Citation - Scopus: 23Cofinitely Weak Supplemented Modules(Taylor and Francis Ltd., 2003) Alizade, Rafail; Büyükaşık, EnginWe prove that a module M is cofinitely weak supplemented or briefly cws (i.e., every submodule N of M with M/N finitely generated, has a weak supplement) if and only if every maximal submodule has a weak supplement. If M is a cws-module then every M-generated module is a cws-module. Every module is cws if and only if the ring is semilocal. We study also modules, whose finitely generated submodules have weak supplements.Article Citation - WoS: 19Citation - Scopus: 19Rings and Modules Characterized by Opposites of Injectivity(Academic Press Inc., 2014) Alizade, Rafail; Büyükaşık, Engin; Er, NoyanIn a recent paper, Aydoǧdu and López-Permouth have defined a module M to be N-subinjective if every homomorphism N→M extends to some E(N)→M, where E(N) is the injective hull of N. Clearly, every module is subinjective relative to any injective module. Their work raises the following question: What is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules, namely injectives? We show, using a dual opposite injectivity condition, that such a ring R is isomorphic to the direct product of a semisimple Artinian ring and an indecomposable ring which is (i) a hereditary Artinian serial ring with J2 = 0; or (ii) a QF-ring isomorphic to a matrix ring over a local ring. Each case is viable and, conversely, (i) is sufficient for the said property, and a partial converse is proved for a ring satisfying (ii). Using the above mentioned classification, it is also shown that such rings coincide with the fully saturated rings of Trlifaj except, possibly, when von Neumann regularity is assumed. Furthermore, rings and abelian groups which satisfy these opposite injectivity conditions are characterized.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 modules
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