Zübeyir Türkoğlu – Page 2 – Dokuz Eylül University Faculty of Science Department of Mathematics

Bialgebroids and Dual Calculus

November 8 2023 by Zübeyir Türkoğlu

Keremcan Doğan , İstanbul Technical University Postdoctoral Researcher.

Date: 10th of November, 2023, Friday.

Time: 12.30 – 13.30.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this talk, we will first explain why algebroids constitute a good framework for generalizations of geometric structures suitable for string and $M$ theories. After a quick overview of their fundamental properties, we will focus on bialgebroids and algebroid calculus. Then, we will be interested in the extensions of Drinfel’d doubles using these notions and certain compatibility conditions between them. We will finish the discussion with the relation between our constructions and exceptional geometries required for string theory. If time permits, we will slightly touch upon the global picture about the formal bundle rackoids.

Tensor Product II

October 30 2023October 30 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 1st of November, 2023, Wednesday, Time: 13:30 – 14:30. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Let $R$ be a ring with identity and let $M$ and $N$ be $R$ -modules ( $M$ a right $R$ -module and $N$ a left R-module). We denote the tensor product with $M\otimes_R N$ . This week, we will continue to talk about various properties of tensor product of M and N. References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.
Bland, P. (2011). Rings and Their Modules. Berlin/New York: de Gruyter.

Tensor Product

October 20 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 25th of October, 2023, Wednesday, Time: 13:30 – 14:30. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Let $R$ be a ring with identity and let $M$ and $N$ be $R$ -modules ( $M$ a right $R$ -module and $N$ a left R-module). In this talk, we will define tensor product of M and N. We will also discuss various properties of tensor product of M and N. We will denote the tensor product with $M\otimes_R N$ .

References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.
Bland, P. (2011). Rings and Their Modules. Berlin/New York: de Gruyter.

Local Rings: Krull-Remak-Schmidt Theorem II

June 5 2023May 29 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 7th of June, 2023, Wednesday, Time: 10:30 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Every injective module over a noetherian ring is a direct sum of directly indecomposable submodules. The question arises as to whether and in what sense such a decomposition is uniquely determined. This question is answered by the Krull-Remak-Schmidt Theorem. The proof of the Krull-Remak-Schmidt Theorem assumes that the endomorphism rings of the direct summands are local rings. Hence we have, first of all, to introduce local rings and then to state sufficient conditions in order that the endomorphism ring of a directly indecomposable module is local. References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.

Local Rings: Krull-Remak-Schmidt Theorem

May 18 2023 by Zübeyir Türkoğlu

Mücahit Bozkurt, Manisa Celal Bayar University. Date: 24th of May, 2023, Wednesday, Time: 10:30 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings). Abstract: Every injective module over a noetherian ring is a direct sum of directly indecomposable submodules. The question arises as to whether and in what sense such a decomposition is uniquely determined. This question is answered by the Krull-Remak-Schmidt Theorem. The proof of the Krull-Remak-Schmidt Theorem assumes that the endomorphism rings of the direct summands are local rings. Hence we have, first of all, to introduce local rings and then to state sufficient conditions in order that the endomorphism ring of a directly indecomposable module is local. References

Kasch, F. (1982). Modules and rings (Vol. 17). Academic press.

The Torsion Free Covers II

May 8 2023 by Zübeyir Türkoğlu

Canan Özeren, Dokuz Eylül University. Date: 10th of May, 2023, Wednesday, Time: 10:30 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: We will continue to talk about the existence and uniqueness (up to isomorphism) torsion-free covers of modules over an integral domain (see [1]). We show that the classical definition of torsion-free cover and the definition of an $F$ -cover, where $F$ is the class of the torsion-free modules, coincide.

References

[1] E. Enochs: Torsion-free covering modules. (1963)

The Torsion Free Covers

May 1 2023April 24 2023 by Zübeyir Türkoğlu

Canan Özeren, Dokuz Eylül University. Date: 03rd of May, 2023, Wednesday, Time: 10:30 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: We talk about the existence and uniqueness (up to isomorphism) torsion-free covers of modules over an integral domain (see [1]). We show that the classical definition of torsion-free cover and the definition of an $F$ -cover, where $F$ is the class of the torsion-free modules, coincide.

References

[1] E. Enochs: Torsion-free covering modules. (1963)

On coGalois Groups III

April 8 2023 by Zübeyir Türkoğlu

Canan Özeren, Dokuz Eylül University. Date: 12th of April, 2023, Wednesday, Time: 10:30 – 12:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings). Abstract:Torsion-free covers exist for abelian groups (see [1]). The coGalois group of a torsion-free cover $\phi: T \rightarrow A$ of an abelian group is defined in [2] as the group of $f: T \rightarrow T$ s.t. $\phi f= \phi$ and is denoted by $G(\phi)$ . The abelian groups for which the coGalois group is trivial were characterized in [3]. The notion of coGalois group can be defined in any category where we have a covering class. In [4], coGalois groups have been studied in the category of representations of the quiver $q_2 : \cdot \rightarrow \cdot$ . We talk about the necessary and sufficient conditions for coGalois group, associated to a torsion free-cover of an object in $(q_2, Z-mod)$ to be trivial. References [1] E. Enochs: Torsion-free covering modules. (1963) [2] E. Enochs, J. R. García Rozas and L. Oyonarte: Compact coGalois groups. (2000). [3] E. Enochs and J. Rada: Abelian groups which have trivial absolute coGalois group. (2005). [4] Paul Hill, Abelian group pairs having a trivial coGalois Group. (2006). [5] Molly Dukun, Phd Thesis [5] Molly Dukun, Phd Thesis

On coGalois Groups II

March 27 2023March 27 2023 by Zübeyir Türkoğlu

Canan Özeren, Dokuz Eylül University.

Date: 29th of March, 2023, Wednesday,

Time: 10:30 – 12:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract:Torsion-free covers exist for abelian groups (see [1]). The coGalois group of a torsion-free cover $\phi: T \rightarrow A$ of an abelian group is defined in [2] as the group of $f: T \rightarrow T$ s.t. $\phi f= \phi$ and is denoted by $G(\phi)$ . The abelian groups for which the coGalois group is trivial were characterized in [3]. The notion of coGalois group can be defined in any category where we have a covering class. In [4], coGalois groups have been studied in the category of representations of the quiver $q_2 : \cdot \rightarrow \cdot$ . We talk about the necessary and sufficient conditions for coGalois group, associated to a torsion free-cover of an object in $(q_2, Z-mod)$ to be trivial.

References

[1] E. Enochs: Torsion-free covering modules. (1963)

[2] E. Enochs, J. R. García Rozas and L. Oyonarte: Compact coGalois groups. (2000).

[3] E. Enochs and J. Rada: Abelian groups which have trivial absolute coGalois group. (2005).

[4] Paul Hill, Abelian group pairs having a trivial coGalois Group. (2006). [5] Molly Dukun, Phd Thesis

[5] Molly Dukun, Phd Thesis

On Rings Whose Finite Length Indecomposable Modules Are Completely Determined by Their Composition Factors III

March 14 2023March 14 2023 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University.

Date: 15th of March, 2023, Wednesday, Time: 10.30 – 12.00.

Place:Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Today we will continue the discussion we started the other day. In particular, we will prove that a commutative Artinian Ring $R$ is of finite representation type if and only if it satisfies $\mathfrak{X}$ , that is, if and only if finitely generated indecomposable $R$ -modules are completely determined by their composition factors.