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TEKİR, ÜNSAL

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Now showing 1 - 10 of 52
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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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    von Neumann regular modules
    (TAYLOR & FRANCIS INC, 2018) TEKİR, ÜNSAL; Jayaram, C.; Tekir, Unsal
    In this paper, we introduce von Neumann regular modules and give many characterizations of von Neumann regular modules. Further, we investigate the relations between von Neumann regular modules and other classical modules. Finally, we characterize Noetherian von Neumann regular modules.
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    PublicationOpen Access
    On the union of graded prime ideals
    (DE GRUYTER OPEN LTD, 2016-01-01) TEKİR, ÜNSAL; Uregen, Rabia Nagehan; Tekir, Unsal; Oral, Kursat Hakan
    In this paper we investigate graded compactly packed rings, which is defined as; if any graded ideal I of R is contained in the union of a family of graded prime ideals of R, then I is actually contained in one of the graded prime ideals of the family. We give some characterizations of graded compactly packed rings. Further, we examine this property on h - Spec(R). We also define a generalization of graded compactly packed rings, the graded coprimely packed rings. We show that R is a graded compactly packed ring if and only if R is a graded coprimely packed ring whenever R be a graded integral domain and h - dim R = 1.
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    PublicationOpen Access
    Second and Secondary Lattice Modules
    (2014) ASLANKARAYİĞİT UĞURLU, EMEL; Çallıalp, Fethi; Tekir, Ünsal; Uğurlu, Emel Aslankarayiğit; Oral, Kürşat Hakan
    Let M be a lattice module over the multiplicative lattice L . A nonzero L -lattice module M is called second if for each a ∈ L , a 1 M = 1 M or a 1 M = 0 M . A nonzero L- lattice module M is called secondary if for each a ∈ L , a 1 M = 1 M or a n 1 M = 0 M for some n > 0 . Our objective is to investigative properties of second and secondary lattice modules.
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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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    Q-modules
    (SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK, 2009) TEKİR, ÜNSAL; Jayaram, C.; Tekir, Uensal
    In this paper we characterize Q-modules and almost Q-modules. Next we estblish some equivalent conditions for an almost Q-module to be a Q-module. Using these results, some characterizations are given for Noetherian Q-modules.
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    PublicationOpen Access
    Prime, weakly prime and almost prime elements in multiplication lattice modules
    (SCIENDO, 2016-01-01) ASLANKARAYİĞİT UĞURLU, EMEL; Ugurlu, Emel Aslankarayigit; Callialp, Fethi; Tekir, Unsal
    In this paper, we study multiplication lattice modules. We establish a new multiplication over elements of a multiplication lattice module. With this multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in multiplication lattice modules.
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    PublicationOpen Access
    On right S-Noetherian rings and S-Noetherian modules
    (TAYLOR & FRANCIS INC, 2018-02-01) TEKİR, ÜNSAL; Bilgin, Zehra; Reyes, Manuel L.; Tekir, Unsal
    In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain S-Noetherian modules with the following consequence in commutative algebra: If a module M over a commutative ring is S-Noetherian with respect to a multiplicative set S that contains no zero-divisors for M, then M has an associated prime.