Person: TEKİR, ÜNSAL
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TEKİR
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ÜNSAL
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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 Metadata only φ-weakly second submodules(2022-10-23) TEKİR, ÜNSAL; Çeken S., Koç S., Tekir Ü.Let R be a commutative ring with identity and M be an R-module. A non-zero submodule N of M is said to be a weakly second submodule if rsN⊆K, where r,s∈R and K is a submodule of M, implies either rN⊆K or sN⊆K. In this paper we introduce and study the concept of φ-weakly second submodules which are generalizations of weakly second submodules. Let φ:S(M)→S(M) be a function where S(M) is the set of all submodules of M. A non-zero submodule N of M is said to be a φ-weakly second submodule if, for any elements a,b of R and a submodule K of M, abN⊆K and abφ(N)⊈K imply either aN⊆K or bN⊆K. We give some properties and characterizations of φ-weakly second submodules and investigate their relationships with weakly second submodules. M is said to be a comultiplication R-module if for every submodule N of M there exists an ideal I of R such that N=(0:M I) where (0:M I)={m∈M:Im=(0)}. We determine φ-weakly second submodules of a comultiplication module. A non-zero submodule N of M is said to be a φ-second submodule if, for any element a of R and a submodule K of M, aN⊆K and aφ(N)⊈K imply either N⊆K or aN=(0). φ-weakly second submodules are also generalizations of φ-second submodules. As a special case we prove that the concept of φ-weakly second submodule coincides with φ-second submodules when M is a comultiplication R-module. Let R=R1×R2, M=M1×M2 where Ri is a ring, Mi is an Ri-module for i=1,2. We investigate the structure of φ-weakly second submodule of the Rmodule M=M1×M2 where M1 and M2 are R-modules.Publication Open Access On normal modules(2022-10-01) TEKİR, ÜNSAL; Jayaram C., Tekir Ü., Koç S., Çeken S.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.Publication Open Access On quasi maximal ideals(2023-12-01) TEKİR, ÜNSAL; Alan M., Kılıç M., Koç S., Tekir Ü.Let R be a commutative ring with 1 ̸= 0. A proper ideal I of R is said tobe a quasi maximal ideal if for every a ∈ R − I, either I + Ra = R or I + Rais a maximal ideal of R. This class of ideals lies between 2-absorbing idealsand maximal ideals which is different from prime ideals. In addition to givefundamental properties of quasi maximal ideals, we characterize principal idealUN-rings with √02= (0), direct product of two fields, and Noetherian zerodimensional modules in terms of quasi maximal ideals.Publication Metadata only Amalgamated algebras along an ideal defined by 1-absorbing-like conditions(2023-01-01) TEKİR, ÜNSAL; El Khalfi A., Kolotoğlu T., Mahdou N., Tekir Ü., Ersoy B. A.Let R be a commutative ring with nonzero identity. A proper ideal I of R is called a 1-absorbing prime ideal (respectively, 1-absorbing primary ideal) if whenever nonunit elements a,b,c∈R with abc∈I, then ab∈I or c∈I (respectively, ab∈I or c∈I–√). The purpose of this paper is to study the transfer of certain 1-absorbing-like properties to amalgamation of A with B along J with respect to f (denoted by A⋈fJ), introduced and studied by D’Anna, Finocchiaro and Fontana. Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties.Publication Metadata only On n-1-absorbing prime ideals(2022-05-01) TEKİR, ÜNSAL; Ulucak G., Koç S., Tekir Ü.In this paper, we introduce and study n -1-absorbing prime ideals of commutative rings. Let R be a ring and n a positive integer. A proper ideal I of R is said to be an n -1-absorbing prime ideal if whenever x1x2…xn+1∈I for some nonunits x1,x2,…,xn+1∈R, then either x1x2…xn∈I or xn+1∈I. It is obvious that 1-1-absorbing (2-1-absorbing) prime ideals are exactly prime (1-absorbing prime) ideals. Various examples and characterizations of n -1-absorbing prime ideals are given.Publication Metadata only On wsq-primary ideals(2023-01-01) ASLANKARAYİĞİT UĞURLU, EMEL; TEKİR, ÜNSAL; Aslankarayiğit Uğurlu E., Tekir Ü., Koç S.We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity and $Q$ a proper ideal of $R$. The proper ideal $Q$ is said to be a weakly strongly quasi-primary ideal if whenever $0\neq ab\in Q$ for some $a,b\in R$, then $a^2\in Q$ or $b\in\sqrt{Q}.$ Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero dimensional rings over which every proper ideal is wsq-primary. Finally, we study finite union of wsq-primary ideals.Publication Metadata only ϕ-s-prime Ideals Of Commutative Rings(2022-10-23) TEKİR, ÜNSAL; Bolat M., Kaya E., Onar S., Koç S., Ersoy B. A. , Tekir Ü.The concept of prime ideals and its generalizations have a distinguished place in commutative algebra since they are not only used in the characterization of various types of rings, but they also have some applications in other areas such as Graph Theory, Cryptology, Topology, Algebraic Geometry, etc. This paper aims to introduce and study ϕ-S-prime ideals of commutative rings which is a new generalization of prime ideals. Let R be a commutative ring with unity, S be a multiplicatively closed subset of R and ϕ:L(R)→L(R)∪{∅} be a function, where L(R) is the lattice of all ideals of R. An ideal I of R is said to be a ϕ-S-prime ideal if there exists a uniform s∈S such that ab∈I-ϕ(I) for some a,b∈R imply that sa∈I or sb∈I. In fact, prime ideals and its many recent generalizations such as S-prime ideals, weakly S-prime ideals and almost S-prime ideals are particular cases of our new concept. In this study, among other things, we determine the relations between ϕ-S-prime ideals and other classical ones. Also, we investigate the behavior of ϕ-S-prime ideals under rings homomorphisms, in factor rings, in quotient rings, in cartesian product of rings, in trivial extension. Finally, as an application of ϕS-prime ideals, we use them to characterize some special rings.Publication Metadata only Commutative graded-S-coherent rings(2023-04-01) TEKİR, ÜNSAL; Assarrar A., Mahdou N., Tekir Ü., Koç S.Publication Open Access Notes on 1-ansorbing prime ideals(2022-01-01) TEKİR, ÜNSAL; Bouba E. M., Tamekkante M., TEKİR Ü., KOÇ S.Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-absorbing prime ideal if xyz is an element of I for some nonunits x, y, z is an element of R, then xy is an element of I or z is an element of I. It is well known that prime ideal double right arrow 1-absorbing prime ideal double right arrow primary ideal double right arrow semi-primary ideal, that is, the class of 1-absorbing prime ideals comes between the classes of prime ideals and primary ideals. Also, the above right arrows are not reversible. In this article, we characterize rings over which every 1-absorbing prime ideal is prime and every primary ideal is 1-absorbing prime. Also, by comparing 1-absorbing prime ideals and other some classical ideals such as 2-absorbing ideals and semi-primary ideals, we characterize Noetherian divided rings and von Neumann regular rings.