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KOÇ, SUAT

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2016 - 201942020 - 20228
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Now showing 1 - 10 of 12
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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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    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 n-absorbing delta-primary ideals
    (SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK, 2018) KOÇ, SUAT; Ulucak, Gulsen; Tekir, Unsal; Koc, Suat
    Let R be a commutative ring with nonzero identity and n be a positive integer. In this paper, we study the concepts of n-absorbing delta-primary ideals and weakly n-absorbing delta-primary ideals, which are the generalizations of delta-primary ideals and weakly delta-primary ideals, respectively. We introduce the concepts of n-absorbing delta-primary ideals and weakly n-absorbing delta-primary ideals. Moreover, we give many properties of these new types of ideals and investigate the relations between these structures.
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    On graded coherent-like properties in trivial ring extensions
    (SPRINGER INT PUBL AG, 2022) KOÇ, SUAT; Assarrar, Anass; Mahdou, Najib; Tekir, Unsal; Koc, Suat
    Let A = circle plus(alpha is an element of G) A(alpha) be a commutative ring with unity graded by an arbitrary grading commutative monoid G, E be a graded A-module and R = A proportional to E the graded trivial extension. In this paper, in the second section, we improve some results on graded trivial extension and we give some new ones and the theme throughout is how homogeneous properties are related to those of A and E. Then, in another section, we introduce and study the notions of graded-v-coherent, graded-quasi-coherent and graded-finite conductor rings, then we study their transfer in the graded trivial extension.
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    On weakly 2-prime ideals in commutative rings
    (TAYLOR & FRANCIS INC, 2021) KOÇ, SUAT; Koc, Suat
    The purpose of the paper is to introduce and study weakly 2-prime ideals in commutative rings. Let A be a commutative ring with a nonzero identity. A proper ideal P of A is said to be a weakly 2-prime ideal if whenever 0 not equal xy is an element of P for some x, y is an element of A, then x(2) is an element of P or y(2) is an element of P: Besides giving various examples and characterizations of weakly 2-prime ideals, we investigate the relations between weakly 2-prime ideals and other classical ideals such as 2-prime ideals, weakly prime ideals and weakly 2-absorbing primary ideals. Furthermore, we study 2-prime avoidance lemma in commutative rings and introduced the class of compactly 2-packed rings. Finally, we investigate the compactly packedness, compactly 2-packedness and coprimely packedness of trivial extension A (sic) M of an A-module M.
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    On S-prime submodules
    (SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK, 2019) KOÇ, SUAT; Sengelen Sevim, Esra; Arabaci, Tarik; Tekir, Unsal; Koc, Suat
    In this study, we introduce the concepts of S-prime submodules and S-torsion-free modules, which are generalizations of prime submodules and torsion-free modules. Suppose S subset of R is a multiplicatively closed subset of a commutative ring R, and let M be a unital R-module. A submodule P of M with (P : (R) M) boolean AND S = empty set is called an S-prime submodule if there is an s is an element of S such that am is an element of P implies sa is an element of(P : (R) M) or sm is an element of P: Also, an R-module M is called S-torsion-free if ann(M) boolean AND S = empty set and there exists s is an element of S such that am = 0 implies sa = 0 or sm = 0 for each a is an element of R and m is an element of M: In addition to giving many properties of S-prime submodules, we characterize certain prime submodules in terms of S-prime submodules. Furthermore, using these concepts, we characterize some classical modules such as simple modules, S-Noetherian modules, and torsion-free modules.
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    On S-multiplication modules
    (TAYLOR & FRANCIS INC, 2020) KOÇ, SUAT; Anderson, Dan D.; Arabaci, Tarik; Tekir, Unsal; Koc, Suat
    In this article, we introduce S-multiplication modules which are a generalization of multiplication modules. Let M be an R-module and a multiplicatively closed subset. M is said to be an S-multiplication module if for each submodule N of M there exist and an ideal I of R such that Besides giving many properties of S-multiplication modules, we generalize some results on multiplication modules to S-multiplication modules. Also, we study S-prime submodules in S-multiplication modules. In particular, we generalize prime avoidance lemma for multiplication modules to S -multiplication modules. Furthermore, we characterize multiplication modules in terms of S-multiplication modules. Communicated by Toma Albu
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    r-Submodules and sr-Submodules
    (SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK, 2018) KOÇ, SUAT; Koc, Suat; Tekir, Unsal
    In this article, we introduce new classes of submodules called r-submodule and special r-submodule, which are two different generalizations of r-ideals. Let M be an R-module, where R is a commutative ring. We call a proper submodule N of M an r-submodule (resp., special r-submodule) if the condition am is an element of N with ann(M) (a) = 0(M) (resp., ann(R)(m) = 0) implies that m E N (resp., a is an element of (N :(R) M)) for each a is an element of R and m is an element of M. We also give various results and examples concerning r-submodules and special r-submodules.
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    Locally torsion-free modules
    (2022-01-01) TEKİR, ÜNSAL; KOÇ, SUAT; Jayaram C., Uǧurlu E. A., TEKİR Ü., KOÇ S.
    © 2023 World Scientific Publishing Company.Recall that a commutative ring R is a locally integral domain if its localization RP is an integral domain for each prime ideal P of R. Our aim in this paper is to extend the notion of locally integral domains to modules. Let R be a commutative ring with a unity and M a nonzero unital R-module. M is called a locally torsion-free module if the localization MP of M is a torsion-free RP-module for each prime ideal P of R. In addition to giving many properties of locally torsion-free modules, we use them to characterize Baer modules, torsion free modules, and von Neumann regular rings.