Fatma Kaynarca

Uniqueness of uniform decompositions in exact categories

Journal of Pure and Applied Algebra, Jun 1, 2021

Abstract Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini ar... more Abstract Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini are extended from abelian categories to weakly idempotent complete exact categories. We give applications to (quasi-)abelian categories, finitely accessible additive categories and exactly definable additive categories.

Almost Projective Modules Over Non-Hereditary Algebras

Contemporary mathematics, 2023

Insertion-Of-Factors-Property on Skew Polynomial Rings

Journal of The Korean Mathematical Society, Nov 1, 2015

In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial ... more In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial rings, introducing the concept of strongly σ-IFP for a ring endomorphism σ. A ring R is said to have strongly σ-IFP if the skew polynomial ring R[x; σ] has IFP. We examine some characterizations and extensions of strongly σ-IFP rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results.

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Rigidness and Extended Armendariz Property

Bulletin of The Korean Mathematical Society, Jan 31, 2011

For a ring endomorphism α of a ring R, Krempa called α a rigid endomorphism if aα(a) = 0 implies ... more For a ring endomorphism α of a ring R, Krempa called α a rigid endomorphism if aα(a) = 0 implies a = 0 for a ∈ R, and Hong et al. called R an α-rigid ring if there exists a rigid endomorphism α. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., α-Armendariz rings and α-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between α-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an α-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.

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Ring Endomorphisms with the Reversible Condition

Communications of The Korean Mathematical Society, Jul 31, 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, b ∈ R. Commutative r... more P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, b ∈ R. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and α an endomorphism of R, we say that R is right (resp., left) α-shifting if whenever aα(b) = 0 (resp., α(a)b = 0) for a, b ∈ R, bα(a) = 0 (resp., α(b)a = 0); and the ring R is called α-shifting if it is both left and right α-shifting. We investigate characterizations of αshifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism α of a ring R, R is right (resp., left) α-shifting if and only if Q(R) is right (resp., left)ᾱ-shifting, whenever there exists the classical right quotient ring Q(R) of R.

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Weak Quasi-Armendariz Rings

Algebra Colloquium, Dec 1, 2011

In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak... more In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/〈xn〉 is weak quasi-Armendariz, where R[x] is the polynomial ring over R and 〈xn〉 is the ideal of R[x] generated by xn. Various properties of weak quasi-Armendariz rings are also observed.

On Skew Polynomial Rings

In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial ... more In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial rings, introducing the concept of strongly σ-IFP for a ring endomorphism σ. A ring R is said to have strongly σ-IFP if the skew polynomial ring R[x;σ] has IFP. We examine some characterizations and extensions of strongly σ-IFP rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results.

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Skew Hurwitz Polinom Halkası Terslenebilir Olan Halkalar ve Genişlemeleri

Afyon Kocatepe University Journal of Sciences and Engineering

Anahtar kelimeler Terslenebilir halka; Skew Hurwitz polinom halkası; Zayıf-terslenebilir halka;-k... more Anahtar kelimeler Terslenebilir halka; Skew Hurwitz polinom halkası; Zayıf-terslenebilir halka;-katı halka Öz Bu çalışmada; bir halkanın terslenebilir olması kavramı, o halka üzerine kurulan skew Hurwitz polinomlar halkasına genişletilerek, zayıf skew Hurwitz terslenebilir (kısaca zayıf-terslenebilir) halka kavramı tanıtılmıştır. Bu halka sınıfının bazı özelikleri ve diğer halka sınıflarıyla ilişkileri incelenmiştir. Zayıf-terslenebilir halkaların bazı genişlemelerinin de zayıf-terslenebilir olup olmadığı araştırılmıştır. Özel olarak, karakteristiği 0 olan bir halkası ve 'nin bir monomorfizması için 'nin zayıf-terslenebilir olması için gerek ve yeter koşul (,) Jordan genişlemesinin zayıf terslenebilir olması gerektiği ispatlanmıştır.

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Extensions of uniserial modules

Rendiconti del Seminario Matematico della Università di Padova, 2020

Let R be any ring and let 0 → A → B → C → 0 be an exact sequence of R-modules which does not spli... more Let R be any ring and let 0 → A → B → C → 0 be an exact sequence of R-modules which does not split with A and C uniserial. Then either B is indecomposable or B has a decomposition of the form B = B1 ⊕ B2 where B1 and B2 are indecomposable and at least one of them is uniserial. Mathematics Subject Classification (2010).Primary: 16D10; Secondary: 16G20.

Uniqueness of uniform decompositions in exact categories

Journal of Pure and Applied Algebra, 2021

Abstract Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini ar... more Abstract Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini are extended from abelian categories to weakly idempotent complete exact categories. We give applications to (quasi-)abelian categories, finitely accessible additive categories and exactly definable additive categories.

The isomorphism problem for uniserial modules over an arbitrary ring

Communications in Algebra, 2020

Firstly, we give a partial solution to the isomorphism problem for uniserial modules of finite le... more Firstly, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules over an arbitrary ring. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers.

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Reversibility of skew Hurwitz series rings

Hacettepe Journal of Mathematics and Statistics, 2020

We study the reversibility of skew Hurwitz series at zero as a generalization of an $\alpha$-rigi... more We study the reversibility of skew Hurwitz series at zero as a generalization of an $\alpha$-rigid ring, introducing the concept of skew Hurwitz reversibility. A ring $R$ is called skew Hurwitz reversible ($SH$-reversible, for short), if the skew Hurwitz series ring $(HR,\alpha)$ is reversible i.e. whenever skew Hurwitz series $f, g\in (HR,\alpha)$ satisfy $fg=0$, then $gf=0$. We examine some characterizations and extensions of $SH$-reversible rings in relation with several ring theoretic properties which have roles in ring theory.

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Rigidness and Extended Armendariz Property

Bulletin of the Korean Mathematical Society, 2011

For a ring endomorphism α of a ring R, Krempa called α a rigid endomorphism if aα(a) = 0 implies ... more For a ring endomorphism α of a ring R, Krempa called α a rigid endomorphism if aα(a) = 0 implies a = 0 for a ∈ R, and Hong et al. called R an α-rigid ring if there exists a rigid endomorphism α. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., α-Armendariz rings and α-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between α-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an α-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.

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On Commutativity of Skew Polynomials at Zero

Bulletin of the Korean Mathematical Society, 2017

We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an ... more We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an α-rigid ring, introducing the concept of strongly skew reversibility. A ring R is be said to be strongly α-skew reversible if the skew polynomial ring R[x; α] is reversible. We examine some characterizations and extensions of strongly α-skew reversible rings in relation with several ring theoretic properties which have roles in ring theory.

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Insertion-Of-Factors-Property on Skew Polynomial Rings

Journal of the Korean Mathematical Society, 2015

In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial ... more In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial rings, introducing the concept of strongly σ-IFP for a ring endomorphism σ. A ring R is said to have strongly σ-IFP if the skew polynomial ring R[x; σ] has IFP. We examine some characterizations and extensions of strongly σ-IFP rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results.

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Ring Endomorphisms with the Reversible Condition

Communications of the Korean Mathematical Society, 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, b ∈ R. Commutative r... more P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, b ∈ R. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and α an endomorphism of R, we say that R is right (resp., left) α-shifting if whenever aα(b) = 0 (resp., α(a)b = 0) for a, b ∈ R, bα(a) = 0 (resp., α(b)a = 0); and the ring R is called α-shifting if it is both left and right α-shifting. We investigate characterizations of αshifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism α of a ring R, R is right (resp., left) α-shifting if and only if Q(R) is right (resp., left)ᾱ-shifting, whenever there exists the classical right quotient ring Q(R) of R.

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On Non-Δ-M-Cosingular Completely ⊕-Δm-Supplemented Modules

RING THEORY 2007 - Proceedings of the Fifth China–Japan–Korea Conference, 2008

ABSTRACT

Weak Quasi-Armendariz Rings

Algebra Colloquium, 2011

In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak... more In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/〈xn〉 is weak quasi-Armendariz, where R[x] is the polynomial ring over R and 〈xn〉 is the ideal of R[x] generated by xn. Various properties of weak quasi-Armendariz rings are also observed.

Skew Hurwitz Polinom Halkası Terslenebilir Olan Halkalar ve Genişlemeleri

Fen ve mühendislik bilimleri dergisi, Dec 31, 2020

Anahtar kelimeler Terslenebilir halka; Skew Hurwitz polinom halkası; Zayıf-terslenebilir halka;-k... more Anahtar kelimeler Terslenebilir halka; Skew Hurwitz polinom halkası; Zayıf-terslenebilir halka;-katı halka Öz Bu çalışmada; bir halkanın terslenebilir olması kavramı, o halka üzerine kurulan skew Hurwitz polinomlar halkasına genişletilerek, zayıf skew Hurwitz terslenebilir (kısaca zayıf-terslenebilir) halka kavramı tanıtılmıştır. Bu halka sınıfının bazı özelikleri ve diğer halka sınıflarıyla ilişkileri incelenmiştir. Zayıf-terslenebilir halkaların bazı genişlemelerinin de zayıf-terslenebilir olup olmadığı araştırılmıştır. Özel olarak, karakteristiği 0 olan bir halkası ve 'nin bir monomorfizması için 'nin zayıf-terslenebilir olması için gerek ve yeter koşul (,) Jordan genişlemesinin zayıf terslenebilir olması gerektiği ispatlanmıştır.

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