Derya Keskin K Tütüncü - Academia.edu (original) (raw)
Papers by Derya Keskin K Tütüncü
In this paper, firstly we show that for lifting modules M and N, M is N-projective if and only if... more In this paper, firstly we show that for lifting modules M and N, M is N-projective if and only if M is epi-N-projective and im-small N projective. Secondly we show that for any weakly supplemented module N, if M \oplus N is small epi-N-projective then M is N-projective.
In [5] and [6], we have introduced a couple of relative generalized epi-projectivities and given ... more In [5] and [6], we have introduced a couple of relative generalized epi-projectivities and given several properties of these projectivities. In this paper, we consider relative generalized injectivities that are dual to these relative projectivities and apply them to the study of direct sums of extending modules. Firstly we prove that for an extending module N, a module M is N-injective if and only if M is mono-Ninjective and essentially N-injective. Then we define a mono-ojectivity that plays an important role in the study of direct sums of extending modules. The structure of (mono-)ojectivity is complicated and hence it is difficult to determine whether these injectivities are inherited by finite direct sums and direct summands even in the case where each module is quasi-continuous. Finally we give several characterizations of these injectivities and find necessary and sufficient conditions for the direct sums of extending modules to be extending
A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g ... more A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g : N → X and any homomorphism f : M → X, there exist decompositions M = M1 ⊕ M2, N = N1 ⊕ N2, a homomorphism h1 : M1 → N1 and an epimorphism h2 : N2 → M2 such that g ◦ h1 = f|M1 and f ◦ h2 = g|N2 . This relative projectivity is very useful for the study on direct sums of lifting modules (cf. [5], [7]). In the definition, it should be noted that we may often consider the case when f to be an epimorphism. By this reason, in this paper we define relative (strongly) generalized epi-projective modules and show several results on this generalized epi-projectivity. We apply our results to the known problem when finite direct sums M1⊕· · ·⊕Mn of lifting modules Mi (i = 1, · · · , n) is lifting.</p
Journal of Algebra and Its Applications
We investigate relative CS-Baer objects in abelian categories in relationship with other relevant... more We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.
Rendiconti del Seminario Matematico della Università di Padova, 2022
We introduce and investigate (dual) relative split objects with respect to a fully invariant shor... more We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to direct sums and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module and comodule categories.
Let R be a P.I.-ring and M any R-module. If M is fully idempotent, then M is a V -module.
Kyungpook Mathematical Journal, 2021
In this paper we investigate the Baer-Kaplansky theorem for module classes on algebras of finite ... more In this paper we investigate the Baer-Kaplansky theorem for module classes on algebras of finite representation types over a field. To do this we construct finite dimensional quiver algebras over any field.
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.
Journal of Algebra and Its Applications, 2020
We introduce and study relatively divisible and relatively flat objects in exact categories in th... more We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.
Taiwanese Journal of Mathematics, 2007
Let R be any ring and let M be any right R-module. M is called hollow-lifting if every submodule ... more Let R be any ring and let M be any right R-module. M is called hollow-lifting if every submodule N of M such that M/N is hollow has a coessential submodule that is a direct summand of M. We prove that every amply supplemented hollow-lifting module with finite hollow dimension is lifting. It is also shown that a direct sum of two relatively projective hollowlifting modules is hollow-lifting.
Journal of Algebra Combinatorics Discrete Structures and Applications, 2017
For a unitary right module M , there are two known partitions of simple modules in the category σ... more For a unitary right module M , there are two known partitions of simple modules in the category σ[M ]: the first one divides them into M-injective modules and M-small modules, while the second one divides them into M-projective modules and M-singular modules. We study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper classes.
We say that a module M is weak lifting if M is supplemented and every supplement submodule of M i... more We say that a module M is weak lifting if M is supplemented and every supplement submodule of M is a direct summand. The module M is called lifting, if it is weak lifting and amply supplemented. This paper investigates the structure of weak lifting modules and lifting modules having small radical over commutative noetherian rings. In this note all
Bulletin of Mathematical Sciences, 2012
We study the ADS * modules which are the dualizations of ADS modules studied by Alahmadi et al. (... more We study the ADS * modules which are the dualizations of ADS modules studied by Alahmadi et al. (J Algebra 352:215-222, 2012). Mainly we prove that an amply supplemented module M is ADS * if and only if M 1 and M 2 are mutually projective whenever M = M 1 ⊕ M 2 if and only if for any direct summand S 1 and a submodule S 2 with M = S 1 + S 2 , the epimorphism α i : M −→ S i /(S 1 ∩ S 2) with Ker(α i) = S j (i = j = 1, 2) can be lifted to an idempotent endomorphism β i of M with β i (M) ⊆ S i .
Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics
Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if for every submodule A... more Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if for every submodule A of M there exists a direct summand D of M such that (A+D)/D⊆τ(M/D) and (A+D)/A⊆τ(M/A). Let τ be a cohereditary preradical. Firstly, for a duo module M=M 1 ⊕M 2 we prove that M is τ-H-supplemented if and only if M 1 and M 2 are τ-H-supplemented. Secondly, let M=⨁ i=1 n M i be a τ-supplemented module. Assume that M i is τ-M j -projective for all j>i. If each M i is τ-H-supplemented, then M is τ-H-supplemented. We also investigate the relations between τ-H-supplemented modules and τ-(⊕-)supplemented modules.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2013
Let M be a right R-module. We call M Rad-D12, if for every sub- module N of M, there exist a dire... more Let M be a right R-module. We call M Rad-D12, if for every sub- module N of M, there exist a direct summand K of M and an epimor- phism α : K → M/N such that Kererα ⊆ Rad(K). We show that a direct summand of a Rad-D12 module need not be a Rad-D12 module. We investigate completely Rad-D12 modules (modules for which every direct summand is a Rad-D12 module). We also show that a direct sum of Rad-D12 modules need not be a Rad-D12 module. Then we deal with some cases of direct sums of Rad-D12 modules.
Vietnam Journal of Mathematics, 2013
Israel Journal of Mathematics, 2014
International Journal of Mathematics, 2005
In this note, we introduce the (small, pseudo-)ℬ(M,X)-cojective modules and we generalize (small,... more In this note, we introduce the (small, pseudo-)ℬ(M,X)-cojective modules and we generalize (small, pseudo-)cojective modules via the class ℬ(M,X). Let M = M1 ⊕ M2 be an X-amply supplemented module with the finite internal exchange property. Then for every decomposition of M = Mi ⊕ Mj, Mi is ℬ(Mj,X)-cojective for i ≠ j, M1 and M2 are X-lifting if and only if M is X-lifting. We also prove that for an X-amply supplemented module M = M1 ⊕ M2 such that M1 and M2 are indecomposable X-lifting modules, if M2 is ℬ(M1,X)-cojective and M1 is small-ℬ(M2,X)-cojective then M is X-lifting.
In this paper, firstly we show that for lifting modules M and N, M is N-projective if and only if... more In this paper, firstly we show that for lifting modules M and N, M is N-projective if and only if M is epi-N-projective and im-small N projective. Secondly we show that for any weakly supplemented module N, if M \oplus N is small epi-N-projective then M is N-projective.
In [5] and [6], we have introduced a couple of relative generalized epi-projectivities and given ... more In [5] and [6], we have introduced a couple of relative generalized epi-projectivities and given several properties of these projectivities. In this paper, we consider relative generalized injectivities that are dual to these relative projectivities and apply them to the study of direct sums of extending modules. Firstly we prove that for an extending module N, a module M is N-injective if and only if M is mono-Ninjective and essentially N-injective. Then we define a mono-ojectivity that plays an important role in the study of direct sums of extending modules. The structure of (mono-)ojectivity is complicated and hence it is difficult to determine whether these injectivities are inherited by finite direct sums and direct summands even in the case where each module is quasi-continuous. Finally we give several characterizations of these injectivities and find necessary and sufficient conditions for the direct sums of extending modules to be extending
A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g ... more A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g : N → X and any homomorphism f : M → X, there exist decompositions M = M1 ⊕ M2, N = N1 ⊕ N2, a homomorphism h1 : M1 → N1 and an epimorphism h2 : N2 → M2 such that g ◦ h1 = f|M1 and f ◦ h2 = g|N2 . This relative projectivity is very useful for the study on direct sums of lifting modules (cf. [5], [7]). In the definition, it should be noted that we may often consider the case when f to be an epimorphism. By this reason, in this paper we define relative (strongly) generalized epi-projective modules and show several results on this generalized epi-projectivity. We apply our results to the known problem when finite direct sums M1⊕· · ·⊕Mn of lifting modules Mi (i = 1, · · · , n) is lifting.</p
Journal of Algebra and Its Applications
We investigate relative CS-Baer objects in abelian categories in relationship with other relevant... more We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.
Rendiconti del Seminario Matematico della Università di Padova, 2022
We introduce and investigate (dual) relative split objects with respect to a fully invariant shor... more We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to direct sums and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module and comodule categories.
Let R be a P.I.-ring and M any R-module. If M is fully idempotent, then M is a V -module.
Kyungpook Mathematical Journal, 2021
In this paper we investigate the Baer-Kaplansky theorem for module classes on algebras of finite ... more In this paper we investigate the Baer-Kaplansky theorem for module classes on algebras of finite representation types over a field. To do this we construct finite dimensional quiver algebras over any field.
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.
Journal of Algebra and Its Applications, 2020
We introduce and study relatively divisible and relatively flat objects in exact categories in th... more We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.
Taiwanese Journal of Mathematics, 2007
Let R be any ring and let M be any right R-module. M is called hollow-lifting if every submodule ... more Let R be any ring and let M be any right R-module. M is called hollow-lifting if every submodule N of M such that M/N is hollow has a coessential submodule that is a direct summand of M. We prove that every amply supplemented hollow-lifting module with finite hollow dimension is lifting. It is also shown that a direct sum of two relatively projective hollowlifting modules is hollow-lifting.
Journal of Algebra Combinatorics Discrete Structures and Applications, 2017
For a unitary right module M , there are two known partitions of simple modules in the category σ... more For a unitary right module M , there are two known partitions of simple modules in the category σ[M ]: the first one divides them into M-injective modules and M-small modules, while the second one divides them into M-projective modules and M-singular modules. We study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper classes.
We say that a module M is weak lifting if M is supplemented and every supplement submodule of M i... more We say that a module M is weak lifting if M is supplemented and every supplement submodule of M is a direct summand. The module M is called lifting, if it is weak lifting and amply supplemented. This paper investigates the structure of weak lifting modules and lifting modules having small radical over commutative noetherian rings. In this note all
Bulletin of Mathematical Sciences, 2012
We study the ADS * modules which are the dualizations of ADS modules studied by Alahmadi et al. (... more We study the ADS * modules which are the dualizations of ADS modules studied by Alahmadi et al. (J Algebra 352:215-222, 2012). Mainly we prove that an amply supplemented module M is ADS * if and only if M 1 and M 2 are mutually projective whenever M = M 1 ⊕ M 2 if and only if for any direct summand S 1 and a submodule S 2 with M = S 1 + S 2 , the epimorphism α i : M −→ S i /(S 1 ∩ S 2) with Ker(α i) = S j (i = j = 1, 2) can be lifted to an idempotent endomorphism β i of M with β i (M) ⊆ S i .
Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics
Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if for every submodule A... more Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if for every submodule A of M there exists a direct summand D of M such that (A+D)/D⊆τ(M/D) and (A+D)/A⊆τ(M/A). Let τ be a cohereditary preradical. Firstly, for a duo module M=M 1 ⊕M 2 we prove that M is τ-H-supplemented if and only if M 1 and M 2 are τ-H-supplemented. Secondly, let M=⨁ i=1 n M i be a τ-supplemented module. Assume that M i is τ-M j -projective for all j>i. If each M i is τ-H-supplemented, then M is τ-H-supplemented. We also investigate the relations between τ-H-supplemented modules and τ-(⊕-)supplemented modules.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2013
Let M be a right R-module. We call M Rad-D12, if for every sub- module N of M, there exist a dire... more Let M be a right R-module. We call M Rad-D12, if for every sub- module N of M, there exist a direct summand K of M and an epimor- phism α : K → M/N such that Kererα ⊆ Rad(K). We show that a direct summand of a Rad-D12 module need not be a Rad-D12 module. We investigate completely Rad-D12 modules (modules for which every direct summand is a Rad-D12 module). We also show that a direct sum of Rad-D12 modules need not be a Rad-D12 module. Then we deal with some cases of direct sums of Rad-D12 modules.
Vietnam Journal of Mathematics, 2013
Israel Journal of Mathematics, 2014
International Journal of Mathematics, 2005
In this note, we introduce the (small, pseudo-)ℬ(M,X)-cojective modules and we generalize (small,... more In this note, we introduce the (small, pseudo-)ℬ(M,X)-cojective modules and we generalize (small, pseudo-)cojective modules via the class ℬ(M,X). Let M = M1 ⊕ M2 be an X-amply supplemented module with the finite internal exchange property. Then for every decomposition of M = Mi ⊕ Mj, Mi is ℬ(Mj,X)-cojective for i ≠ j, M1 and M2 are X-lifting if and only if M is X-lifting. We also prove that for an X-amply supplemented module M = M1 ⊕ M2 such that M1 and M2 are indecomposable X-lifting modules, if M2 is ℬ(M1,X)-cojective and M1 is small-ℬ(M2,X)-cojective then M is X-lifting.