Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article On Purities Relative To Minimal Right Ideals(Pleiades Publishing, 2023) Alagöz, Yusuf; Alizade, Rafail; Büyükaşık, Engin; Sağbaş, SelçukAbstract: We call a right module M weakly neat-flat if (Formula presented.) is surjective for any epimorphism (Formula presented.) and any simple right ideal S . A left module M is called weakly absolutely s-pure if (Formula presented.) is monic, for any monomorphism (Formula presented.) and any simple right ideal S . These notions are proper generalization of the neat-flat and the absolutely s-pure modules which are defined in the same way by considering all simple right modules of the ring, respectively. In this paper, we study some closure properties of weakly neat-flat and weakly absolutely s-pure modules, and investigate several classes of rings that are characterized via these modules. The relation between these modules and some well-known homological objects such as projective, flat, injective and absolutely pure are studied. For instance, it is proved that R is a right Kasch ring if and only if every weakly neat-flat right R -module is neat-flat (moreover if R is right min-coherent) if and only if every weakly absolutely s-pure left R -module is absolutely s-pure. The rings over which every weakly neat-flat (resp. weakly absolutely s-pure) module is injective and projective are exactly the QF rings. Finally, we study enveloping and covering properties of weakly neat-flat and weakly absolutely s-pure modules. The rings over which every simple right ideal has an epic projective envelope are characterized. © 2023, Pleiades Publishing, Ltd.Article Citation - WoS: 12Citation - Scopus: 11Poor Modules With No Proper Poor Direct Summands(Academic Press Inc., 2018) Alizade, Rafail; Büyükaşık, Engin; López-Permouth, Sergio; Yang, LiuAs 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.Article Citation - WoS: 3Citation - Scopus: 3Co-Coatomically Supplemented Modules(Springer Verlag, 2017) Alizade, Rafail; Güngör, SerpilIt is shown that if a submodule N of M is co-coatomically supplemented and M/N has no maximal submodule, then M is a co-coatomically supplemented module. If a module M is co-coatomically supplemented, then every finitely M-generated module is a co-coatomically supplemented module. Every left R-module is co-coatomically supplemented if and only if the ring R is left perfect. Over a discrete valuation ring, a module M is co-coatomically supplemented if and only if the basic submodule of M is coatomic. Over a nonlocal Dedekind domain, if the torsion part T(M) of a reduced module M has a weak supplement in M, then M is co-coatomically supplemented if and only if M/T (M) is divisible and TP (M) is bounded for each maximal ideal P. Over a nonlocal Dedekind domain, if a reduced module M is co-coatomically amply supplemented, then M/T (M) is divisible and TP (M) is bounded for each maximal ideal P. Conversely, if M/T (M) is divisible and TP (M) is bounded for each maximal ideal P, then M is a co-coatomically supplemented module.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.Article Citation - WoS: 3Citation - Scopus: 3Small Supplements, Weak Supplements and Proper Classes(Hacettepe Üniversitesi, 2016) Alizade, Rafail; Büyükaşık, Engin; Durğun, YılmazLet SS denote the class of short exact sequences E:0 → Af→ 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 such that 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 class SS coincides with the smallest proper class containing the class of short exact sequences determined by weak supplement submodules. The homological objects, such as, SS-projective and SScoinjective modules are investigated. In order to describe the class SS, we investigate small supplemented modules, i.e. the modules each of whose submodule has a small supplement. Besides proving some closure properties of small supplemented modules, we also give a complete characterization of these modules over Dedekind domains.Article Citation - WoS: 7Citation - Scopus: 7The Proper Class Generated by Weak Supplements(Taylor and Francis Ltd., 2014) Alizade, Rafail; Demirci, Yılmaz Mehmet; Durğun, Yılmaz; Pusat, DilekWe 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.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.Article Citation - WoS: 11Citation - Scopus: 11Cofinitely Supplemented Modular Lattices(Springer Verlag, 2011) Alizade, Rafail; Toksoy, Sultan EylemIn 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 v b there exists an element c of b/0 such that a v c 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.Conference Object Resolutions in Cotorsion Theories(American Institute of Physics, 2010) Akıncı, Karen; Alizade, RafailWe consider the λ- (μ-) and λ̄- (μ̄-) dimensions of modules taken under a cotorsion theory (F, C) satisfying the Hereditary Condition, and establish some inequalities between the dimensions of the modules of a short exact sequence, not necessarily Hom (F, -) exact. We investigate the question of whether the property of having a (special) F- or C-resolution of length n is resolving, closed under extensions or coresolving and establish some inequalities connecting the λ- (μ-) and λ̄- (μ̄-) dimensions of modules in a short exact sequence. © 2010 American Institute of Physics.Article Citation - WoS: 14Citation - Scopus: 14Cofinitely Weak Supplemented Lattices(Indian National Science Academy, 2009) Alizade, Rafail; Toksoy, Sultan EylemIn this paper it is shown that an E-complemented complete modular lattice L with small radical is weakly supplemented if and only if it is semilocal. L is a cofinitely weak supplemented lattice if and only if every maximal element of L has a weak supplement in L. If )α/0 is a cofinitely weak supplemented (weakly supplemented) sublattice and 1/α has no maximal element (1/α is weakly supplemented and a has a weak supplement in L), then L is cofinitely weak supplemented (weakly supplemented).