Person: KOÇ, SUAT
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KOÇ
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SUAT
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Publication Open 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.Publication Open Access On phi-1-absorbing prime ideals(SPRINGER HEIDELBERG, 2021-12) KOÇ, SUAT; Yildiz, Eda; Tekir, Unsal; Koc, SuatIn 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.Publication Open Access Quasi regular modules and trivial extension(HACETTEPE UNIV, FAC SCI, 2020-12-31) KOÇ, SUAT; Jayaram, Chillumuntala; Tekir, Unsal; Koc, SuatRecall 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.Publication Open Access On weakly 1-absorbing prime ideals(SPRINGER-VERLAG ITALIA SRL, 2021-03-14) KOÇ, SUAT; Koc, Suat; Tekir, Unsal; Yildiz, EdaThis 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.Publication Open Access On (m, n)-semiprime submodules(ESTONIAN ACAD PUBLISHERS, 2021) KOÇ, SUAT; Pekin, Ayten; Koc, Suat; Ugurlu, Emel AslankarayigitThis 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.Publication Open 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.Publication Open Access On Strongly ??-regular Modules(2020-08-01) KOÇ, SUAT; Suat KOÇIn this article, we introduce the notion of strongly ??-regular module which is a generalization_x000D_ of von Neumann regular module in the sense [13]. Let ?? be a commutative ring with 1 ≠ 0 and_x000D_ ?? a multiplication ??-module. ?? is called a strongly ??-regular module if for each ?? ∈ ??,_x000D_ (????)m = ???? = ??2?? for some ?? ∈ ?? and ?? ∈ ℕ. In addition to give many properties and_x000D_ examples of strongly ??-regular modules, we also characterize certain class of modules such as_x000D_ von Neumann regular modules and second modules in terms of this new class of modules. Also,_x000D_ we determine when the localization of any family of submodules at a prime ideal commutes_x000D_ with the intersection of this family._x000D_ _x000D_ Keywords: von Neumann regular module, (??, ??)-closed ideal, strongly ??-regular module,_x000D_ Krull dimension, (∗)-property, localization.Publication Open Access On 1-absorbing delta-primary ideals(OVIDIUS UNIV PRESS, 2021-11-01) KOÇ, SUAT; El Khalfi, Abdelhaq; Mahdou, Najib; Tekir, Unsal; Koc, SuatLet R be a commutative ring with nonzero identity. Let J(R) be the set of all ideals of R and let delta : J(R) - -> J(R) be a function. Then delta is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J subset of I, we have L subset of delta(L) and delta(J) subset of delta(I). Let delta be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of delta-primary ideals. A proper ideal I of R is said to be a 1-absorbing delta-primary ideal if whenever nonunit elements a, b, c is an element of R and abc is an element of I, then ab is an element of I or c is an element of delta(I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing delta-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.Publication Open Access On S-2-absorbing submodules and vn-regular modules(OVIDIUS UNIV PRESS, 2020-07-01) KOÇ, SUAT; Ulucak, Gulsen; Tekir, Unsal; Koc, SuatLet R be a commutative ring and M an R-module. In this article, we introduce the concept of S-2-absorbing submodule. Suppose that S subset of R is a multiplicatively closed subset of R. A submodule P of M with (P :(R) M) boolean AND S = empty set is said to be an S-2-absorbing submodule if there exists an element s is an element of S and whenever abm is an element of P for some a, b is an element of R and m is an element of M, then sab is an element of (P :(R) M) or sam is an element of P or sbm is an element of P. Many examples, characterizations and properties of S-2-absorbing submodules are given. Moreover, we use them to characterize von Neumann regular modules in the sense [9].