algebra

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)

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

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

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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

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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.

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

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Victor Blasco Jimenez, Dokuz Eylül University.

Date: 1st 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: From the Fundamental Theorem of Abelian Groups we can deduce that every finite length indecomposable module over \mathbb{Z} is uniquely determined by its composition factors set, meaning that if we have two indecomposable finite length abelian groups N^1 and N^2 with the same length and, if for any 0\subseteq N_{1}^j\subseteq ... \subseteq N_{r}^j=N^j composition series for N^j, j=1,2, we get \{N_{i+1}^1/N_{i}^1\}_{i=0}^{r-1}=\{N_{i+1}^2/N_{i}^2\}_{i=0}^{r-1}, then we must have N^1\cong N^2. In this series of talks , we will study this property about the finite length indecomposable abelian groups in a more general way. We will start by focusing on the class of commutative rings R which satisfy it, showing that it contains the class of Dedekind Domains. If time permits, we will see that if R is any unital ring (not necessarily commutative) satisfying this property, which we will call “Property \mathfrak{X}“, and I is an ideal of R, then also R/I satisfies it. This work is part of my ongoing master thesis “Some methods of Category Theory in the Representation Theory of Artin Algebras”.

The Auslander-Reiten Quiver of the 2-Kronecker Algebra III

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İrem Yıldız, Dokuz Eylül University.
Date: 22nd of February, 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: The Auslander-Reiten quiver of a finite dimensional k-algebra A is the quiver whose vertices are the isoclasses of indecomposable modules over A, and the number of arrows between two vertices [M],[N] is given by the k-dimension of Irr_{A}(M,N).
In this talk, we will review some basic facts about almost split sequences and explain briefly how to construct them. Our aim is to construct the Auslander-Reiten quiver of the path algebra associated to the 2-Kronecker quiver.

The Auslander-Reiten Quiver of the 2-Kronecker Algebra II

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İrem Yıldız, Dokuz Eylül University.
Date: 17th of January, 2023, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: The Auslander-Reiten quiver of a finite dimensional k-algebra A is the quiver whose vertices are the isoclasses of indecomposable modules over A, and the number of arrows between two vertices [M],[N] is given by the k-dimension of Irr_{A}(M,N).
In this talk, we will review some basic facts about almost split sequences and explain briefly how to construct them. Our aim is to construct the Auslander-Reiten quiver of the path algebra associated to the 2-Kronecker quiver.