Yazar "Alagoz, Yusuf" seçeneğine göre listele

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    An alternative perspective on pure-projectivity of modules
    (Springer International Publishing Ag, 2020) Alagoz, Yusuf; Durgun, Yilmaz
    The study of pure-projectivity is accessed from an alternative point of view. Given modules M and N, M is said to be N-pure-subprojective if for every pure epimorphism g : B -> N and homomorphism f : M -> N, there exists a homomorphism h : M -> B such that gh = f. For a module M, the pure-subprojectivity domain of M is defined to be the collection of all modules N such that M is N-pure-subprojective. We obtain characterizations for various types of rings and modules, including FP-injective and FP-projective modules, von Neumann regular rings and pure-semisimple rings in terms of pure-subprojectivity domains. As pure-subprojectivity domains clearly include all pure-projective modules, a reasonable opposite to pure-projectivity in this context is obtained by considering modules whose pure-subprojectivity domain consists of only pure-projective. We refer to these modules as psp-poor. Properties of pure-subprojectivity domains and of psp-poor modules are studied.
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    Homological objects of min-pure exact sequences
    (Hacettepe Univ, Fac Sci, 2024) Alagoz, Yusuf; Moradzadeh-Dehkordi, Ali
    In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are considered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring R is two-sided K & ouml;the if all right R-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with J(R)(2) = 0, it is proven that all R-modules are min-pure projective if and only if R is either a field or a quasi-Frobenius ring of composition length 2.
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    Max-projective modules
    (World Scientific Publ Co Pte Ltd, 2021) Alagoz, Yusuf; Buyukasik, Engin
    Weakening the notion of R-projectivity, a right R-module M is called max-projective provided that each homomorphism f : M -> R/I, where I is any maximal right ideal, factors through the canonical projection pi : R -> R/I. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are R-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring R, we prove that injective modules are R-projective if and only if R = A x B, where A is QF and B is a small ring. If R is right hereditary and right Noetherian then, injective right modules are max-projective if and only if R = S x T, where S is a semisimple Artinian and T is a right small ring. If R is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective.
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    On max-flat and max-cotorsion modules
    (Springer, 2021) Alagoz, Yusuf; Buyukasik, Engin
    In this paper, we continue to study and investigate the homological objects related to s-pure and neat exact sequences of modules and module homomorphisms. A right module A is called max-flat if Tor(1)(R) (A, R/I) = 0 for any maximal left ideal I of R. A right module B is said to be max-cotorsion if Ext(R)(1)(A, B) = 0 for any max-flat right module A. We characterize some classes of rings such as perfect rings, max-injective rings, SF rings and max-hereditary rings by max-flat and max-cotorsion modules. We prove that every right module has a max-flat cover and max-cotorsion envelope. We show that a left perfect right max-injective ring R is QF if and only if maximal right ideals of R are finitely generated. The max-flat dimensions of modules and rings are studied in terms of right derived functors of -circle times-. Finally, we study the modules that are injective and flat relative to s-pure exact sequences.
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    On MF-projective modules
    (Hacettepe Univ, Fac Sci, 2021) Alagoz, Yusuf
    In this paper, we study the left orthogonal class of max-flat modules which are the homological objects related to s-pure exact sequences of modules and module homomorphisms. Namely, a right module A is called MF-projective if Ext(R)(1) (A, B) = 0 for any max-flat right R-module B, and A is called strongly MF-projective if Ext(R)(1 )(A, B) = 0 for all max-flat right R-modules B and all i >= 1. Firstly, we give some properties of MF-projective modules and SMF-projective modules. Then we introduce and study MF-projective dimensions for modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. We characterize some classes of rings such as perfect rings, QF rings and max-hereditary rings by (S)MF-projective modules. We also study the rings whose right ideals are MF-projective. Finally, we characterize the rings whose MF-projective modules are projective.
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    On minimal absolutely pure domain of RD-fllat modules
    (Tubitak Scientific & Technological Research Council Turkey, 2022) Alagoz, Yusuf
    Given modules A(R) and B-R, B-R is called absolutely A(R)-pure if for every extension C-R of B-R, A circle times B -> A circle times C is a monomorphism. The class (Fl) under bar (-1)(A(R)) ={B-R : B-R is absolutely A(R)-pure} is called the absolutely pure domain of a module A(R). If B-R is divisible, then all short exact sequences starting with B is RD-pure, whence B is absolutey A-pure for every RD-flat module A(R). Thus the class of divisible modules is the smallest possible absolutely pure domain of an RD-flat module. In this paper, we consider RD-flat modules whose absolutely pure domains contain only divisible modules, and we referred to these RD-flat modules as rd-indigent. Properties of absolutely pure domains of RD-flat modules and of rd-indigent modules are studied. We prove that every ring has an rd-indigent module, and characterize rd-indigent abelian groups. Furthermore, over (commutative) SRDP rings, we give some characterizations of the rings whose nonprojective simple modules are rd-indigent.
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    On Purities Relative to Minimal Right Ideals
    (Maik Nauka/Interperiodica/Springer, 2023) Alagoz, Yusuf; Alizade, Rafail; Buyukasik, Engin; Sagbas, Selcuk
    We call a right module M weakly neat-flat if Hom(S, N) -> Hom(S, M) is surjective for any epimorphism N -> M and any simple right ideal S. A left module M is called weakly absolutely s-pure if S circle times M -> S circle times N is monic, for any monomorphism M -> N 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.
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    On rings admitting nonzero homomorphisms between non-projective modules
    (Taylor & Francis Inc, 2023) Alagoz, Yusuf; Durgun, Yilmaz; Izci, Kemal
    In a recent paper, Turkoglu have studied the rings admitting nonzero homomorphisms between any two non-projective modules (property (T)). His work raises the following question: What are the rings with nonzero maps between non-projective right modules and singular simple modules? As a generalization of the property (T), we consider two families of rings: those rings with nonzero maps from any non-projective module to any singular simple module (property (T1)) and those rings with nonzero maps from any singular simple module to any non-projective module (property (T2)). Complete characterizations of both classes of rings are obtained and it is shown, in particular, that the rings satisfying property (T) are precisely those rings satisfying both properties (T1) and (T2). We show that R satisfies (T) if and only if R has a unique singular simple right R-module, and R is either two-sided Artinian hereditary serial or right completely coretractable. Furthermore, we prove that a commutative ring R has the property (T) if and only if there is a ring decomposition R approximately equal to AxB , where A is semisimple Artinian and B satisfies one of the following conditions: (1) B is a semi-Artinian max ring with unique singular simple right R-module. (2) B is an Artinian hereditary serial ring with unique singular simple right R-module.
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    On simple-injective modules
    (World Scientific Publ Co Pte Ltd, 2023) Alagoz, Yusuf; Benli-Goral, Sinem; Buyukasik, Engin
    For a right module M, we prove that M is simple-injective if and only if M is min-N-injective for every cyclic right module N. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is QF if and only if simple-injective right modules are projective. For a commutative Noetherian ring R, we prove that every finitely generated simple-injective R-module is projective if and only if R = A x B, where A is QF and B is hereditary. An abelian group is simple-injective if and only if its torsion part is injective. We show that the notions of simple-injective, strongly simple-injective, soc-injective and strongly soc-injective coincide over the ring of integers.
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    On Singly Flat and Singly Injective Modules
    (Springer Singapore Pte Ltd, 2021) Alagoz, Yusuf
    Recall that a left module A (resp. right module B) is said to be singly injective (resp. singly flat) if Ext(R)(1) (F/K, A) = 0 (resp. Tor(1)(R)(B, F/K) = 0) for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we continue to study and investigate the homological objects related to singly flat and singly injective modules and module homomorphisms. Along the way, the right orthogonal class of singly flat right modules and the left orthogonal class of singly injective left modules are introduced and studied. These concepts are used to extend the some known results and to characterize pseudo-coherent rings and left singly injective rings. In terms of some derived functors, some homological dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and left PP rings are given. Finally, we study the singly flatness and singly injectivity of homomorphism modules over a commutative ring.
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    Pure-Direct-Injective Modules
    (Maik Nauka/Interperiodica/Springer, 2022) Maurya, Sanjeev Kumar; Das, Soumitra; Alagoz, Yusuf
    In this paper, we study the class of modules having the property that if any pure submodule is isomorphic to a direct summand of such a module then the pure submodule is itself a direct summand. These modules are termed as pure-direct-injective modules (or pure-C2 modules). We have characterized the rings whose pure-C2 modules satisfy certain conditions, such as being C2, (pure-) injective, projective, or dual-Utumi. For instance, it is proved that if R is a right Noetherian ring over which every pure-C2 right R-module is pure-injective, then R is Kroll-Schmidt semiperfect. The rings over which every pure-C2 module is injective, projective and dual-Utumi are exactly the semisimple rings. Also, it is shown that a ring R is right perfect if and only if every projective right R- module is pure-C2.
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    RELATIVE SUBCOPURE-INJECTIVE MODULES
    (Ankara Univ, Fac Sci, 2020) Alagoz, Yusuf
    In this paper, copure-injective modules are examined from an alternative perspective. For two modules A and B, A is called B-subcopure-injective if for every copure monomorphism f : B -> C and homomorphism g : B -> A, there exists a homomorphism h : C -> A such that hf = g. The class CPJ(-1) (A) = {B : A is B-subcopure-injective} is called the subcopure-injectivity domain of A. We obtain characterizations of copure-injective modules, right CDS rings and right V-rings with the help of subcopure-injectivity domains. Since subcopure-injectivity domains clearly contains all copure-injective modules, studying the notion of modules which are subcopure-injective only with respect to the class of copure-injective modules is reasonable. We refer to these modules as sc-indigent. We studied the properties of subcopure-injectivity domains and of sc-indigent modules and investigated these modules over some certain rings.
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    Rings whose nonsingular right modules are R-projective
    (Charles Univ, Fac Mathematics & Physics, 2021) Alagoz, Yusuf; Benli, Sinem; Buyukasik, Engin
    A right R-module M is called R-projective provided that it is projective relative to the right R-module R-R. This paper deals with the rings whose all nonsingular right modules are R-projective. For a right nonsingular ring R, we prove that R-R is of finite Goldie rank and all nonsingular right R-modules are R-projective if and only if R is right finitely Sigma-CS and fiat right R-modules are R-projective. Then, R-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that R-projectivity of nonsingular injective right modules is equivalent to R-projectivity of the injective hull E(R-R). In this case, the injective hull E(R-R) has the decomposition E(R-R) = U-R circle plus V-R, where U is projective and Hom(V, R/I) = 0 for each right ideal I of R. Finally, we focus on the right orthogonal class N-perpendicular to of the class IV of nonsingular right modules.