Tensor Product II

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

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

Tensor Product

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

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

Local Rings: Krull-Remak-Schmidt Theorem II

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

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

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

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

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

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

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.

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

Victor Blasco Jimenez, Dokuz Eylül University. Date: 8th 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 show that a commutative Artinian Ring R of finite representation type satisfies \mathfrak{X}, that is, finitely generated indecomposable R-modules are completely determined by their composition factors.