Browsing by Author "Alizade, Rafail"

Now showing 1 - 17 of 17

Abelian groups whose nonzero endomorphisms have nonessential kernels
(WORLD SCIENTIFIC PUBL CO PTE LTD, 2017) Rafail Alizade; Surajo Ibrahim Isah; Alizade, Rafail; Isah, Surajo Ibrahim
In this paper we describe completely the K-singular subgroup of an abelian group and a K-nonsingular abelian group in terms of the basic subgroups of its p-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a K- nonsingular abelian group are K-nonsingular and give a condition under which a pure extension of a K-nonsingular abelian group by a K-nonsingular group is K-nonsingular.
Closures of proper classes
(University of Miskolc matronto@uni-miskolc.hu, 2016) Refail Alizade; Yılmaz Mehmet Demirci; Alizade, Rafail; Demirci, Yilmaz Mehmet
For an integral domain R we consider the closures M (Mr r ε R) of a submodule M of an R-module N consisting of elements n of N with tn 2 M (rmn ε M) for some nonzero t ε R (m ε Z+) and its connections with usual closure M of M in N. Using these closures we study the closures P and Pr of a proper class P of short exact sequences and give a decomposition for the class of quasi-splitting short exact sequences of abelian groups into the direct sum of "p-closures" of the class Split of splitting short exact sequences and description of closures of some classes. In the general case of an arbitrary ring we generalize these closures of a proper class P by means of homomorphism classes F and G and prove that under some conditions this closure is a propier classes. © 2020 Elsevier B.V. All rights reserved.
Citation - WoS: 3
Citation - Scopus: 1
⊕-co-coatomically supplemented and co-coatomically semiperfect modules
(HACETTEPE UNIV FAC SCI, 2018) Rafail Alizade; Serpil Gungor; Alizade, Rafail; Gungor, Serpil
In this paper it is shown that a factor module of an circle plus-co-coatomically supplemented module is not in general circle plus-co-coatomically supplemented. If M is circle plus-co-coatomically supplemented and U is a fully invariant submodule of M then M/U is circle plus-co-coatomically supplemented. A ring R is left perfect if and only if R-(N) is an circle plus-co-coatomically supplemented R-module. A projective module M is co-coatomically semiperfect if and only if M is circle plus-co-coatomically supplemented. A ring is semiperfect if and only if every finitely generated free R-module is co-coatomically semiperfect.
Citation - WoS: 11
Citation - Scopus: 12
Cofinitely Supplemented Modular Lattices
(SPRINGER HEIDELBERG, 2011) Rafail Alizade; Sultan Eylem Toksoy; Alizade, Rafail; Toksoy, Sultan Eylem
In this paper it is shown that a lattice L is a cofinitely supplemented lattice if and only if every maximal element of L has a supplement in L. If a/0 is a cofinitely supplemented sublattice and 1/a has no maximal element then L is cofinitely supplemented. A lattice L is amply cofinitely supplemented if and only if every maximal element of L has ample supplements in L if and only if for every cofinite element a and an element b of L with a boolean OR b = 1 there exists an element c of b/0 such that a boolean OR c = 1 where c is the join of finite number of local elements of b/0. In particular a compact lattice L is amply supplemented if and only if every maximal element of L has ample supplements in L.
Güvercin yuvası ilkesi uygulamaları
(2015) Al, Yunus; Kocakaya, Salih; Alizade, Rafail; Alızada, Rafaıl
In this thesis we Show how to apply the pigeonhole principle for solving various mathematical olympiad problems. Furthermore we give the solution of the problems on pigeonhole principle submitted in the first round of the TÜBİTAK mathematical olympiads. We hope that this thesis will be helpfull for high school students that take part in the mathematical olympiads.
İndirgemeli diziler ve uygulamaları
(2014) Sağbaş, Selçuk; Alizade, Rafail
Bu çalışmada; indirgemeli dizilerin bazı sayma problemlerinin çözümündeki uygulamaları ile ilgili kullanılışı gösterildi. Sayma problemlerinde indirgeme bağıntısının kuruluşu anlatıldı. İndirgeme bağıntıları başlıklar halinde sınıflandırıldı. Ayrıca bazı örneklerde indirgeme bağıntısı kullanılarak dizilerin genel terimlerinin bulunması ile ilgili çeşitli yöntemler olduğu üzerinde duruldu. Bu yöntemler teleskopik, karakteristik ve üretici fonksiyon olarak incelendi. Uygulama bölümünde ise indirgemeli diziler ve sayma problemleri ile ilgili sorular çözüldü.
Inequality Olympiad Problems
(2016) Başdaş, Abdulsamet; Alizade, Rafail
Bu tez esas otarak altı bölümden oluşmaktadır, Birinci bölümde, tezimiz ile ilgili anlaşılırlığı sağlamak için ön bilgiler verilmiştir, ikinci bölümde., Ortalama ile ilgili eşitsizlikler ve kendi aralarındaki ilişkiden bahsettik, Ayrıca, ortalamalarla ilgili uluslararası Olimpiyat soruları yazıp, her soruya düzgün bir veya birden fazla ,çözüm yolu getirdik, üçüncü bölümde Cauchy-Schwarz eşitsizliği, Dördüncü bölümde Holder eşitsizliği, Beşininci bölümde Chebyshev eşitsizliğini ve alt başlık olarak da Yeniden Düzenleme eşitsizligini, Altıncı bölümde Ball Geornetrik eşitsilliklere ve bunlarla ilgili Ulusal ve Uluslararası olimpiyat sorularına özgün çözümler yapıldı.
Kare kalanlar
(2014) Sağlam, Alpaslan; Alizade, Rafail
Bu tezde ikinci dereceden bir denkliğin çözümü için kare kalanlar kullanılması incelenmiştir. Sayıların değişik mod'larda kare kalan olup olmadığının araştırılması için Euler Kriterinin, Gauss Lemmasının, Karesel Karşılık for-mülünün ve Legendre sembolünün özelliklerinin kullanılmıştır. Bu sonuçlar sayı teorisi ile ilgili matematik olimpiyat sorularının çözümü için kullanılmıştır.
Citation - WoS: 5
Citation - Scopus: 8
Modules and abelian groups with minimal (pure-) projectivity domains
(WORLD SCIENTIFIC PUBL CO PTE LTD, 2017) Rafail Alizade; Damla Dede Sipahi; Alizade, Rafail; Sipahi, Damla Dede
In this paper we give a complete description of the projectively poor abelian groups and prove that there exists a pure projectively poor abelian group. We show that over a commutative Artinian ring every module having a projectively poor factor module by a pure submodule is itself projectively poor. We also give some other properties of pure projectively poor modules.
Citation - WoS: 2
Citation - Scopus: 2
On rings with one middle class of injectivity domains
(Udruga Matematicara Osijek, 2022) Refail Alizade; Yılmaz Mehmet Demirci; Burcu Nişanci Türkmen; Ergül Türkmen; Türkmen, Burcu Nişancı; Alizade, Rafail; Türkmen, Ergül; Demirci, Yilmaz Mehmet
A module M is said to be modest if the injectivity domain of M is the class of all crumbling modules. In this paper we investigate the basic properties of modest modules. We provide characterizations of some classes of rings using modest modules. In particular we show that a ring having the class of crumbling modules as the only right middle class of injectivity domains is either a right V-ring or right Noetherian, and a commutative ring with this property is regular. We also give criteria for a ring having the class of crumbling modules as the only right middle class of injectivity domains. © 2022 Elsevier B.V. All rights reserved.
Citation - WoS: 13
Citation - Scopus: 14
Poor and pi-poor Abelian groups
(Taylor and Francis Inc. 325 Chestnut St Suite 800 Philadelphia PA 19106, 2017) Refail Alizade; Engi̇n İ. Büyükaşik; Alizade, Rafail; Büyükaşık, Engİn
In 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. © 2016 Elsevier B.V. All rights reserved.
Citation - WoS: 12
Citation - Scopus: 11
Poor modules with no proper poor direct summands
(Academic Press Inc. apjcs@harcourt.com, 2018) Refail Alizade; Engi̇n İ. Büyükaşik; Sergio R. López-Permouth; Liu Yang; López-Permouth, Sergio R.; Büyükaşık, Engİn; Alizade, Rafail; Yang, Liu
As a mean to provide intrinsic characterizations of poor modules the notion of a pauper module is introduced. A module is a pauper if it is poor and has no proper poor direct summand. We show that not all rings have pauper modules and explore conditions for their existence. In addition we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity: one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule. The second condition holds for the ring of integers and is just as significant as the first one for Noetherian rings since in that context modules having poor pure submodules must themselves be poor. It is shown that the existence of paupers is equivalent to the Noetherian condition for rings with no middle class. As indecomposable poor modules are pauper we study rings with no indecomposable right middle class (i.e. the ring whose indecomposable right modules are pauper or injective). We show that semiartinian V-rings satisfy this property and also that a commutative Noetherian ring R has no indecomposable middle class if and only if R is the direct product of finitely many fields and at most one ring of composition length 2. Structure theorems are also provided for rings without indecomposable middle class when the rings are Artinian serial or right Artinian. Rings for which not having an indecomposable middle class suffices not to have a middle class include commutative Noetherian and Artinian serial rings. The structure of poor modules is completely determined over commutative hereditary Noetherian rings. Pauper Abelian groups with torsion-free rank one are fully characterized. © 2018 Elsevier B.V. All rights reserved.
Citation - WoS: 3
Citation - Scopus: 1
Pure-direct-projective modules
(WORLD SCIENTIFIC PUBL CO PTE LTD, 2024) Rafail Alizade; Sultan Eylem Toksoy; Alizade, Rafail; Toksoy, Sultan Eylem
In this paper we introduce and study the pure-direct-projective modules that is the modules M every pure submodule A of which with M/A isomorphic to a direct summand of M is a direct summand of M. We characterize rings over which every right R-module is pure-direct-projective. We examine for which rings or under what conditions pure-direct-projective right R-modules are direct-projective projective quasi-projective pure-projective flat or injective. We prove that over a Noetherian ring every injective module is pure-direct-projective and a right hereditary ring R is right Noetherian if and only if every injective right R-module is pure-direct-projective. We obtain some properties of pure-direct-projective right R-modules which have DPSP and DPIP.
Citation - WoS: 19
Citation - Scopus: 19
Rings and modules characterized by opposites of injectivity
(ACADEMIC PRESS INC ELSEVIER SCIENCE, 2014) Rafail Alizade; Engin Buyukasik; Noyan Er; Er, Noyan; Alizade, Rafail; Buyukasik, Engin
In a recent paper Aydogdu and Lopez-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 J(2) = 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.
Citation - WoS: 3
Citation - Scopus: 3
Small supplements weak supplements and proper classes
(Hacettepe Univ, FAC Sci, 2016) Engin BÜYÜKAŞIK; Rafail ALİZADE; Yılmaz DUR?GUN; Buyukasik, Engin; Durgun, Yilmaz; Alizade, Rafail; Dur?gun, Yılmaz
Let SS denote the class of short exact sequences E :0 ->, A->, B ->,C ->, 0 of R-modules and R-module homomorphisms such that f (A)has a small supplement in B i.e. there exists a submodule K of M suchthat f (A) + K = B and f (A) ? K is a small module. It is shown that SS is a proper class over left hereditary rings. Moreover in this case the proper classSS coincides with the smallest proper class containing the class of short exact sequences determined by weak supplementsubmodules. The homological objects such as SS-projective and SScoinjective modules are investigated. In order to describe the classSS we investigate small supplemented modules i.e. the modules each ofwhose submodule has a small supplement. Besides proving some closure properties of small supplemented modules we also give a completecharacterization of these modules over Dedekind domains
Citation - WoS: 7
Citation - Scopus: 6
Test modules for flatness
(Universita di Padova, 2017) Refail Alizade; Yılmaz Durǧun; Alizade, Rafail; Durğun, Yılmaz
A right R-module M is said to be a test module for flatness (shortly: an f-test module) provided for each left R-module N Tor(M N) = 0 implies N is flat. f-test modules are a flat version of the Whitehead test modules for injectivity defined by Trlifaj. In this paper the properties of f-test modules are investigated and are used to characterize various families of rings. The structure of a ring over which every (finitely generated) right R-module is flat or f-test is investigated. Abelian groups that are Whitehead test modules for injectivity or f-test are characterized. © 2023 Elsevier B.V. All rights reserved.
Citation - WoS: 7
Citation - Scopus: 7
THE PROPER CLASS GENERATED BY WEAK SUPPLEMENTS
(TAYLOR & FRANCIS INC, 2014) Rafail Alizade; Yilmaz M. Demirci; Yilmaz Durgun; Dilek Pusat; Pusat, Dilek; Durǧun, Yilmaz; Alizade, Rafail; Demirci, Yilmaz M.
We show that for hereditary rings the smallest proper classes containing respectively the classes of short exact sequences determined by small submodules submodules that have supplements and weak supplement submodules coincide. Moreover we show that this class can be obtained as a natural extension of the class determined by small submodules. We also study injective projective coinjective and coprojective objects of this class. We prove that it is coinjectively generated and its global dimension is at most 1. Finally we describe this class for Dedekind domains in terms of supplement submodules.