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Citation - WoS: 2
Citation - Scopus: 1
The Difference of Hyperharmonic Numbers Via Geometric and Analytic Methods
(Korean Mathematical Society, 2022) Altuntaş, Çağatay; Göral, Haydar; Sertbaş, Doğa Can
Our 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.
Citation - WoS: 1
Citation - Scopus: 1
Applications of Class Numbers and Bernoulli Numbers To Harmonic Type Sums
(Korean Mathematical Society, 2021) Göral, Haydar; Sertbaş, Doğa Can
Divisibility 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.
Rings and Modules Characterized by Opposites of Fp-Injectivity
(Korean Mathematical Society, 2019) Büyükaşık, Engin; Kafkas Demirci, Gizem
Let R be a ring with unity. Given modules M-R and N-R, M-R is said to be absolutely N-R-pure if M circle times N -> L circle times N is a monomorphism for every extension L-R of M-R. For a module M-R, the subpurity domain of M-R is defined to be the collection of all modules N-R such that M-R is absolutely N-R-pure. Clearly M-R is absolutely F-R-pure for every flat module F-R, and that M-R is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, M-R is said to be a test for flatness by subpurity (or t.f.b.s. for short) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. Every ring has a right t.f.b.s. module. R-R is t.f.b.s. and every finitely generated right ideal is finitely presented if and only if R is right semihereditary. A domain R is Priifer if and only if R is t.f.b.s. The rings whose simple right modules are t.f.b.s. or injective are completely characterized. Some necessary conditions for the rings whose right modules are t.f.b.s. or injective are obtained.
Citation - WoS: 1
Citation - Scopus: 1
Reidemeister Torsion and Orientable Punctured Surfaces
(Korean Mathematical Society, 2018) Dirican, Esma; Sözen, Yaşar
Let Σg,n,b denote the orientable surface obtained from the closed orientable surface Σg of genus g ≥ 2 by deleting the interior of n ≥ 1 distinct topological disks and b ≥ 1 points. Using the notion of symplectic chain complex, the present paper establishes a formula for computing Reidemeister torsion of the surface Σg,n,b in terms of Reidemeister torsion of the closed surface Σg, Reidemeister torsion of disk, and Reidemeister torsion of punctured disk.
Citation - WoS: 2
Citation - Scopus: 3
On W-Local Modules and Rad-Supplemented Modules
(Korean Mathematical Society, 2014) Büyükaşık, Engin; Tribak, Rachid
All modules considered in this note are over associative commutative rings with an identity element. We show that a w-local module M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that w-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated amply Rad-supplemented left modules over any ring (not necessarily commutative).
Citation - WoS: 8
Citation - Scopus: 8
Coneat Submodules and Coneat-Flat Modules
(Korean Mathematical Society, 2014) Büyükaşık, Engin; Durgun, Yılmaz
A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism N → S can be extended to a homomorphism M → S. M is called coneat-flat if the kernel of any epimorphism Y → M → 0 is coneat in Y. It is proven that (1) coneat submodules of any right R-module are coclosed if and only if R is right K-ring; (2) every right R-module is coneat-flat if and only if R is right V -ring; (3) coneat submodules of right injective modules are exactly the modules which have no maximal submodules if and only if R is right small ring. If R is commutative, then a module M is coneatflat if and only if M+ is m-injective. Every maximal left ideal of R is finitely generated if and only if every absolutely pure left R-module is m- injective. A commutative ring R is perfect if and only if every coneat-flat module is projective. We also study the rings over which coneat-flat and flat modules coincide.

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