Yazar "Ceken, Secil" seçeneğine göre listele

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    Comultiplication modules relative to a hereditary torsion theory
    (Taylor & Francis Inc, 2019) Ceken, Secil
    Let R be a commutative ring with identity and tau be a hereditary torsion theory on R-Mod. In this article, we introduce and study the concept of tau-comultiplication module. We present several properties and characterizations of tau-comultiplication modules. We also investigate modules for which every submodule has a unique tau-pseudo-complement and prove that every tau-comultiplication module is a module with unique tau-pseudo-complements.
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    Dual Zariski Spaces of Modules
    (World Scientific Publ Co Pte Ltd, 2023) Ceken, Secil
    Let R be a commutative ring with identity, M be an R-module, L (M ) denote the set of all submodules of M and G subset of L ( M) \ { 0(M) } . For any submodule N of M, we set GV(d) ( N) = { K is an element of G : K subset of N } and G zeta(d) (M ) = { GV(d) ( N) : N is an element of L (M ) } . Consider chi subset of L ( R) \ { R } , where L (R ) is the set of all ideals of R. We set chi V (I ) = { J is an element of chi : I subset of J } and chi zeta (R ) = { chi V (I ) : I is an element of L (R ) } for any ideal I of R. In this paper, we investigate when, for arbitrary chi and G as above, chi zeta (R ) and G zeta(d) (M ) form a topology and a semimodule, respectively. We investigate the structure of G zeta(d) (M ) in the case that it is a semimodule.
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    Generalizations of strongly hollow ideals and a corresponding topology
    (Amer Inst Mathematical Sciences-Aims, 2021) Ceken, Secil; Yuksel, Cem
    In this paper, we introduce and study the notions of M-strongly hollow and M-PS-hollow ideals where M is a module over a commutative ring R. These notions are generalizations of strongly hollow ideals. We investigate some properties and characterizations of M-strongly hollow (M-PS-hollow) ideals. Then we define and study a topology on the set of all M-PS-hollow ideals of a commutative ring R. We investigate when this topological space is irreducible, Noetherian, T-0, T-1 and spectral space.
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    MODULES AND THE SECOND CLASSICAL ZARISKI TOPOLOGY
    (Univ Studi Catania, Dipt Matematica, 2018) Ceken, Secil; Alkan, Mustafa
    Let R be an associative ring with identity and Spec(s)(M) denote the set of all second submodules of a right R-module M. In this paper, we present a number of new results for the second classical Zariski topology on Spec(s)(M) for a right R-module M. We obtain a characterization of semisimple modules by using the second spectrum of a module. We prove that if R is a ring such that every right primitive factor of R is right artinian, then every non-zero submodule of a second right R-module M is second if and only if M is a fully prime module. We give some equivalent conditions for Spec(s)(M) to be a Hausdorff space or T-i-space when the right R-module M has certain algebraic properties. We obtain characterizations of commutative Quasi-Frobenius and artinian rings by using topological properties of the second classical Zariski topology. We give a full characterization of the irreducible components of Spec(s)(M) for a non-zero injective right module M over a ring R such that every prime factor of R is right or left Goldie.
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    On graded 2-absorbing and graded weakly 2-absorbing ideals
    (Hacettepe Univ, Fac Sci, 2019) Al-Zoubi, Khaldoun; Abu-Dawwas, Rashid; Ceken, Secil
    In this paper, we introduce and study graded 2-absorbing and graded weakly 2-absorbing ideals of a graded ring which are different from 2-absorbing and weakly 2-absorbing ideals. We give some properties and characterizations of these ideals and their homogeneous components. We investigate graded (weakly) 2-absorbing ideals of R-1 x R-2 where R-1 and R-2 are two graded rings.
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    On normal modules
    (Taylor & Francis Inc, 2023) Jayaram, Chillumuntala; Tekir, Unsal; Koc, Suat; Ceken, Secil
    Recall that a commutative ring R is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module M is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules.
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    On S-second spectrum of a module
    (Springer-Verlag Italia Srl, 2022) Ceken, Secil
    Let R be a commutative ring with identity, S be a multiplicatively closed subset of R. A submodule N of an R-module M with ann(R)(N) boolean AND S = empty set is called an S-second submodule of M if there exists a fixed s is an element of S, and whenever rN subset of K, where r is an element of R and K is a submodule of M, then either rsN = 0 or sN subset of K. The set of all S-second submodules of M is called S-second spectrum of M and denoted by S-Specs (M). In this paper, we construct and study two topologies on S-Spec(s) (M). We investigate some connections between algebraic properties of M and topological properties of S-Spec(s) (M) such as seperation axioms, compactness, connectedness and irreducibility.
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    On Strongly 2-Absorbing Second Submodules
    (Amer Inst Physics, 2018) Ceken, Secil; Alkan, Mustafa
    In this paper, we study on the concept of strongly 2-absorbing second submodule which is a dual notion of 2-absorbing submodule and a generalization of second submodule. We give some properties and characterizations of this submodule class and investigate the relationships with second and secondary submodules.
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    On the interior of a submodule with respect to a set of ideals
    (Natl Inquiry Services Centre Pty Ltd, 2019) Ceken, Secil
    In this paper, we investigate interior operations on submodules and introduce a new interior operation by using a certain submodule class. Let R be a commutative ring with identity and be a set of ideals of R. We define -second submodules and -interior of a submodule. We show that second, secondary and strongly second submodules are special types of -second submodules. We investigate several properties of -interiors of submodules and give a concrete expression of - interior of a submodule of an Artinian module. We use the concept of -interior of a submodule to find some results on -second submodules and attached primes of an Artinian module.
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    On the Upper Dual Zariski Topology
    (Univ Nis, Fac Sci Math, 2020) Ceken, Secil
    Let R be a ring with identity and M be a left R-module. The set of all second submodules of M is called the second spectrum of M and denoted by Spec(s)(M). For each prime ideal p of R we define Spec(p)(s)(M) := {S is an element of Spec(s)(M) : ann(R)(S) = p g. A second submodule Q ofMis called an upper second submodule if there exists a prime ideal p of R such that Spec(p)(s)(M)not equal (sic) and Q = Sigma S is an element of Spec(p)(s)(M) S. The set of all upper second submodules ofMis called upper second spectrumofMand denoted by u:Specs(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u:Spec(s)(M) with the dual Zarsiki topology. Also, we topologize u:Specs(M) with the patch topology and the finer patch topology. We show that for every left R-moduleM, u:Spec(s)(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u:Spec(s)(M) is a compact space with the patch and finer patch topology. Finally, by applying Hochster's characterization of a spectral space, we show that if M is an Artinian left R-module, then u:Specs(M) with the dual Zariski topology is a spectral space.
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    A sheaf on the second spectrum of a module
    (Taylor & Francıs Inc, 2018) Ceken, Secil; Alkan, Mustafa
    Let R be a commutative ring with identity and Spec(5)(M) denote the set all second submodules of an R-module M. In this paper, we construct and study a sheaf of modules, denoted by O(N, M), on Spec(5)(M) equipped with the dual Zariski topology of M, where N is an R-module. We give a characterization of the sections of the sheaf O(N, M) in terms of the ideal transform module. We present some interrelations between algebraic properties of N and the sections of O(N, M). We obtain some morphisms of sheaves induced by ring and module homomorphisms.