Year: 2023

Local Rings: Krull-Remak-Schmidt Theorem II

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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

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

Local Rings: Krull-Remak-Schmidt Theorem

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

The Torsion Free Covers II

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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

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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)

+1 Step in Career: Student Clubs

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The theme of our career event this month is +1 Step in Career: Student Clubs, and we host the Computer Science and Artificial Intelligence Club, Google Developer Students Clubs-DEU and Business Student Club in our event. The event is open to all mathematics department students and those who are interested. Those who are interested are asked to contact the moderator (Dr. Zübeyir Türkoğlu).

Speakers:

Date and Time: 17.04.2023, 11:30

Location: https://online.deu.edu.tr/
Channel: Kariyerde +1

On coGalois Groups III

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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 rings whose cyclic modules have cyclic injective hulls

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Prof. Dr. Christian Lomp, Department of Mathematics, University of Porto in Porto, Portugal. Date: 19th of April 2023, Wednesday. Time: 11:00. Place: Online/Microsoft Teams- Meeting ID: 351 128 968 15 Passcode: Gy4n4B

Abstract: In 1964, Barbara Osofsky proved in her PhD thesis that a ring whose cyclic modules are injective is semisimple Artinian. William Cadwell in his PhD thesis from 1966 studied when injective hulls of cyclic modules are cyclic and termed them hypercyclic rings. He characterised left perfect left hypercyclic rings as well as commutative local hypercyclic rings. In this talk we will revise the literature on rings whose cyclic modules have cyclic injective hulls and present some more recent results, obtained jointly with Mohamed Yousif and Yiqiang Zhou.

On coGalois Groups II

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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

Graduate Education in Mathematics

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Assoc. Prof. Dr. Burcu SİLINDİR YANTIR (Vice Chair), Date: 28.03.2023, Time: 16:00, Place: online.deu.edu.tr

In cooperation with Dokuz Eylül University Faculty of Science Department of Mathematics and Career Planning Center, a conversation moderated by Assist.Prof.Dr. Celal Cem SARIOĞLU and on “Graduate Education in Mathematics” will be held with Assoc. Prof. Dr. Burcu SİLİNDİR YANTIR, who is vice chair of Mathematics department.  Those who are interested are asked to contact the moderator.