Mathematics / Matematik
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Article Citation - WoS: 3Citation - Scopus: 3Dual Kasch rings(World Scientific Publishing, 2023) Lomp, Christian; Büyükaşık, Engin; Yurtsever, Haydar BaranIt is well known that a ring R is right Kasch if each simple right R-module embeds in a projective right R-module. In this paper we study the dual notion and call a ring R right dual Kasch if each simple right R-module is a homomorphic image of an injective right R-module. We prove that R is right dual Kasch if and only if every finitely generated projective right R-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field k, a finite dimensional k-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.Article Citation - WoS: 1Citation - Scopus: 1Unique decompositions into w-ideals for strong Mori domains(World Scientific Publishing, 2022) Hamdi, Haleh; Ay Saylam, Başak; Gürbüz, EzgiA commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In [P. Goeters and B. Olberding, Unique decomposition into ideals for Noetherian domains, J. Pure Appl. Algebra 165 (2001) 169-182], the UDI property has been characterized for Noetherian integral domains. In this paper, we aim to study the UDI-like property for strong Mori domains; domains satisfying the ascending chain condition on w-ideals.Article Citation - WoS: 1Citation - Scopus: 1Rank One Perturbations Supported by Hybrid Geometries and Their Deformations(American Institute of Physics, 2022) Erman, Fatih; Seymen, Sema; Turgut, O. TeomanWe study the hybrid type of rank one perturbations in ℝ2 and ℝ3, where the perturbation supported by a circle/sphere is considered together with the delta potential supported by a point outside of the circle/sphere. The construction of a self-adjoint Hamiltonian operator associated with formal expressions for the rank one perturbation supported by a circle and by a point is explicitly given. Bound state energies and scattering properties for each problem are also studied. Finally, we consider the rank one perturbation supported by a deformed circle/sphere and show that the first order change in bound state energies under small deformations of the circle/sphere has a simple geometric interpretation.Article Citation - WoS: 2Citation - Scopus: 1The Difference of Hyperharmonic Numbers Via Geometric and Analytic Methods(Korean Mathematical Society, 2022) Altuntaş, Çağatay; Göral, Haydar; Sertbaş, Doğa CanOur motivation in this note is to find equal hyperharmonic numbers of different orders. In particular, we deal with the integerness property of the difference of hyperharmonic numbers. Inspired by finite-ness results from arithmetic geometry, we see that, under some extra assumption, there are only finitely many pairs of orders for two hyper-harmonic numbers of fixed indices to have a certain rational difference. Moreover, using analytic techniques, we get that almost all differences are not integers. On the contrary, we also obtain that there are infinitely many order values where the corresponding differences are integers.Article Unique decompositions into regular ideals for Marot rings(Taylor & Francis, 2022) Ay Saylam, Başak; Gürbüz, EzgiLet R be a commutative ring. We say that R has the unique decomposition into regular ideals (UDRI) property if, for any R-module which decomposes into a finite direct sum of regular ideals, this decomposition is unique up to the order and isomorphism class of the regular ideals. In this paper, we will prove some preliminary results for Marot rings whose regular ideals are finitely generated and give a necessary and sufficient condition for these rings to satisfy the UDRI property.Article Citation - WoS: 1Citation - Scopus: 1Applications of Class Numbers and Bernoulli Numbers To Harmonic Type Sums(Korean Mathematical Society, 2021) Göral, Haydar; Sertbaş, Doğa CanDivisibility properties of harmonic numbers by a prime number p have been a recurrent topic. However, finding the exact p-adic orders of them is not easy. Using class numbers of number fields and Bernoulli numbers, we compute the exact p-adic orders of harmonic type sums. Moreover, we obtain an asymptotic formula for generalized harmonic numbers whose p-adic orders are exactly one.Article Exact Time-Evolution of a Generalized Two-Dimensional Quantum Parametric Oscillator in the Presence of Time-Variable Magnetic and Electric Fields(American Institute of Physics, 2022) Atılgan Büyükaşık, Şirin; Çayiç, ZehraThe time-dependent Schrodinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei-Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin-Man'ko-Trifonov and the Lewis-Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy-Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields. Published under an exclusive license by AIP Publishing.Article Citation - WoS: 2Citation - Scopus: 2On simple-injective modules(World Scientific Publishing, 2022) Alagöz, Yusuf; Benli Göral, Sinem; Büyükaşık, EnginFor a right module M, we prove that M is simple-injective if and only if M is min-N-injective for every cyclic right module N. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is QF if and only if simple-injective right modules are projective. For a commutative Noetherian ring R, we prove that every finitely generated simple-injective R-module is projective if and only if R = A × B, where A is QF and B is hereditary. An abelian group is simple-injective if and only if its torsion part is injective. We show that the notions of simple-injective, strongly simple-injective, soc-injective and strongly soc-injective coincide over the ring of integers.Article Es-W(Taylor & Francis, 2021) Ay Saylam, Başak; Hamdi, HalehWe introduce and study the notion of ES-w-stability for an integral domain R. A nonzero ideal I of R is called ES-w-stable if (I-2)(w) = (JI)(w) for some t-invertible ideal J of R contained in I, and I is called weakly ES-w-stable if I-w = (JE)(w) for some t-invertible fractional ideal J of R and w-idempotent fractional ideal E of R. We define R to be an ES-w-stable domain (resp., a weakly ES-w-stable domain) if every nonzero ideal of R is ES-w-stable (resp., weakly ES-w-stable). These notions allow us to generalize some well-known properties of ES-stable and weakly ES-stable domains.Article Citation - WoS: 5Citation - Scopus: 5On simple-direct modules(Taylor and Francis Ltd., 2021) Büyükaşık, Engin; Demir, Özlem; Diril, MügeRecently, in a series of papers “simple” versions of direct-injective and direct-projective modules have been investigated. These modules are termed as “simple-direct-injective” and “simple-direct-projective,” respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given. © 2020 Taylor & Francis Group, LLC.