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KOÇ, SUAT

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KOÇ

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2016 - 201942020 - 202217
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    On Divided Modules
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2020) KOÇ, SUAT; Tekir, Unsal; Ulucak, Gulsen; Koc, Suat
    Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where a is an element of R. In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where m is an element of M (r is an element of R). In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension R proportional to M.
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    PublicationOpen Access
    On modules satisfying s-noetherian spectrum condition
    (2022-03-01) KOÇ, SUAT; TEKİR, ÜNSAL; Özen M., Naji O. A., Tekir Ü., Koç S.
    Let R be a commutative ring having nonzero identity and M be a unital R-module. Assume that S ⊆ R is a multiplicatively closed subset of R. Then, M satisfies SNoetherian spectrum condition if for each submodule N of M, there exist s ∈ S and a finitely generated submodule F ⊆ N such that sN ⊆ radM (F), where radM (F) is the prime radical of F in the sense (McCasland and Moore in Commun Algebra 19(5):1327–1341, 1991). Besides giving many properties and characterizations of SNoetherian spectrum condition, we prove an analogous result to Cohen’s theorem for modules satisfying S-Noetherian spectrum condition. Moreover, we characterize modules having Noetherian spectrum in terms of modules satisfying the S-Noetherian spectrum condition.
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    PublicationOpen Access
    On phi-1-absorbing prime ideals
    (SPRINGER HEIDELBERG, 2021-12) KOÇ, SUAT; Yildiz, Eda; Tekir, Unsal; Koc, Suat
    In this paper, we introduce phi-1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity 1 not equal 0 and phi : I(R) -> I(R) boolean OR {theta} be a function where I( R) is the set of all ideals of R. A proper ideal I of R is called a phi-1-absorbing prime ideal if for each nonunits x, y, z is an element of R with xyz is an element of I - phi(I), then either xy is an element of I or z is an element of I. In addition to give many properties and characterizations of phi-1-absorbing prime ideals, we also determine rings in which every proper ideal is phi-1-absorbing prime.
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    PublicationOpen Access
    Quasi regular modules and trivial extension
    (HACETTEPE UNIV, FAC SCI, 2020-12-31) KOÇ, SUAT; Jayaram, Chillumuntala; Tekir, Unsal; Koc, Suat
    Recall that a ring R is said to be a quasi regular ring if its total quotient ring q(R) is von Neumann regular. It is well known that a ring R is quasi regular if and only if it is a reduced ring satisfying the property: for each a is an element of R, ann(R)(ann (R) (a)) = ann(R)(b) for some b is an element of R. Here, in this study, we extend the notion of quasi regular rings and rings which satisfy the aforementioned property to modules. We give many characterizations and properties of these two classes of modules. Moreover, we investigate the (weak) quasi regular property of trivial extension.
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    PublicationOpen Access
    On weakly 1-absorbing prime ideals
    (SPRINGER-VERLAG ITALIA SRL, 2021-03-14) KOÇ, SUAT; Koc, Suat; Tekir, Unsal; Yildiz, Eda
    This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let A be a commutative ring with a nonzero identity 1 not equal 0. A proper ideal P of A is said to be a weakly 1-absorbing prime ideal if for every nonunits x, y, z. A with 0 not equal xyz. P, then xy is an element of P or z is an element of P. In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in C( X), which is the ring of continuous functions of a topological space X.
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    PublicationOpen Access
    On (m, n)-semiprime submodules
    (ESTONIAN ACAD PUBLISHERS, 2021) KOÇ, SUAT; Pekin, Ayten; Koc, Suat; Ugurlu, Emel Aslankarayigit
    This paper aims to introduce a new class of submodules, called (m, n)-semiprime submodule, which is a generalization of semiprime submodule. Let M be a unital A-module and m,n is an element of N. Then a proper submodule P of M is said to be an (m, n)-semiprime submodule if whenever a(m)x is an element of P for some a is an element of A,x is an element of M, then a(n)x is an element of P. In addition to giving many characterizations and properties of this kind of submodules, we also use them to characterize von Neumann regular modules.
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    PublicationOpen Access
    On S-comultiplication modules
    (2022-01-01) KOÇ, SUAT; Yıldız E., Tekir Ü., Koç S.
    Let R be a commutative ring with 1 ̸= 0 and M be an R-module. Suppose that S ⊆ R is a multiplicatively closed set of R. Recently Sevim et al. in [19] introduced the notion of an S -prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules, S -Noetherian modules and etc. Afterwards, in [2], Anderson et al. defined the concepts of S -multiplication modules and S -cyclic modules which are S -versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to S -multiplication and S -cyclic modules. Here, in this article, we introduce and study S -comultiplication modules which are the dual notion of S -multiplication module. We also characterize certain classes of rings/modules such as comultiplication modules, S -second submodules, S -prime ideals and S -cyclic modules in terms of S -comultiplication modules. Moreover, we prove S -version of the dual Nakayama’s Lemma.
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    On graded 1-absorbing prime ideals
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2021) KOÇ, SUAT; Abu-Dawwas, Rashid; Yildiz, Eda; Tekir, Unsal; Koc, Suat
    Let G be a group with identity e and R be a G-graded commutative ring with unity 1. In this article, we introduce and study the concept of graded 1-absorbing prime ideals which is a generalization of graded prime ideals. A proper graded ideal P of R is called graded 1-absorbing prime if for all nonunit elements x,y,z is an element of h(R) such that xy z is an element of P, then either xy is an element of P or z is an element of P.
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    On S-Zariski topology
    (TAYLOR & FRANCIS INC, 2021) KOÇ, SUAT; Yildiz, Eda; Ersoy, Bayram Ali; Tekir, Unsal; Koc, Suat
    Let R be a commutative ring with nonzero identity and, S subset of R be a multiplicatively closed subset. An ideal P of R with P boolean AND S = theta is called an S-prime ideal if there exists an (fixed) s is an element of S and whenver ab is an element of P for a, b is an element of R then either sa is an element of P or sb is an element of P. In this article, we construct a topology on the set Spec(S)(R) of all S-prime ideals of R which is generalization of prime spectrum of R. Also, we investigate the relations between algebraic properties of R and topological properties of Spec(S)(R) like compactness, connectedness and irreducibility.
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    On S-comultiplication modules
    (SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK) KOÇ, SUAT; Yildiz, Eda; Tekir, Unsal; Kuc, Suat
    Let R be a commutative ring with 1 not equal 0 and M be an R-module. Suppose that S subset of R is a multiplicatively closed set of R. Recently Sevim et al. in [19] introduced the notion of an S-prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules, S-Noetherian modules and etc. Afterwards, in [2], Anderson et al. defined the concepts of S-multiplication modules and S-cyclic modules which are S-versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to S-multiplication and S-cyclic modules. Here, in this article, we introduce and study S-comultiplication modules which are the dual notion of S-multiplication module. We also characterize certain classes of rings/modules such as comultiplication modules, S-second submodules, S-prime ideals and S-cyclic modules in terms of S-comultiplication modules. Moreover, we prove S-version of the dual Nakayama's Lemma.