Zübeyir Türkoğlu – Page 4 – Dokuz Eylül University Faculty of Science Department of Mathematics

The Existence of Almost Split Sequences

November 29 2022November 29 2022 by Zübeyir Türkoğlu

Fatma Kaynarca, Afyon Kocatepe University.
Date: 29th of November, 2022, Tuesday, Time: 13.00 – 16.30.
Place: Online-Sakai-Graduate Meetings

Abstract: In this seminar, the existence of almost split sequences in the category of modules built on a finite-dimensional algebra will be proved with a functorial approach.

On Higher Commutators

November 22 2022November 22 2022 by Zübeyir Türkoğlu

M. Pınar Eroğlu, Dokuz Eylül University.
Date: 25th of October, 2022, Tuesday, Time: 13:30 - 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract:  A general result on higher commutators due to Herstein I.N. states that if A is a noncommutative simple algebra over a field of characteristic not 2, then the only higher commutators of A are A and [A,A]. For a more general approach, the question that arises naturally is: Which assumptions should be required to obtain a result similar to above in the case where A is an arbitrary noncommutative unital algebra? In this talk, we characterize higher commutators of unital algebras discussing the question above.

Almost Split and Minimal Morphisms

November 21 2022November 21 2022 by Zübeyir Türkoğlu

Fatma Kaynarca, Afyon Kocatepe University.
Date: 21th of November, 2022, Tuesday, Time: 13.00 – 16.30.
Place: Online-Sakai-Graduate Meetings

Abstract: In this seminar, almost split morphisms, another concept that use to characterize almost split sequences, will be introduced and their relationship to irreducible morphisms and minimal morphisms will be examined.

Almost Split Sequences

November 14 2022November 14 2022 by Zübeyir Türkoğlu

Fatma Kaynarca, Afyon Kocatepe University.
Date: 15th of November, 2022, Tuesday, Time: 13.00 – 14.00.
Place: Online-Sakai-Graduate Meetings

Abstract: Almost split sequences arose from an attempt to understand the morphisms lying in the radical of a module category are minimal non-split short exact sequences. This sequences were introduced by Maurice Auslander and Idun Reitenin 1974-1975 and have become a central tool in the theory of representations of finite dimensional algebras. We start our discussion in seminar with a short description of the radical of a module category. Then we will define and study irreducible morphisms, almost split morphisms, minimal morphisms, almost split sequences and also give some characterizations of these notions.

Modules with chain conditions up to isomorphism II

November 5 2022November 5 2022 by Zübeyir Türkoğlu

Zübeyir Türkoğlu, Dokuz Eylül University.
Date: 8th of November, 2022, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this seminar we will continue to talk about isoartinian, isonoetherian and isosimple modules and rings, see [1]. First we will give characterizations of being right Noetherian ring over semiprime right isoartinian rings. Then we can talk about some open problems. Finally, we will talk about what has been done about these concepts and what we can do.

References

[1] A. Facchini and Z. Nazemian, Modules with chain conditions up to isomorphism, Journal of Algebra, pp. 578–601, Vol. 453, 2016.

Modules with chain conditions up to isomorphism

October 18 2022October 16 2022 by Zübeyir Türkoğlu

Zübeyir Türkoğlu, Dokuz Eylül University.
Date: 18th of October, 2022, Tuesday, Time: 13:30 – 14:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this seminar we will talk about isoartinian, isonoetherian and isosimple modules and rings. These notions have been introduced by Facchini and Nazemian in [1]. Let $R$ be a ring. A right $R$ -module $M$ is called isoartinian (isonoetherian) if, for every descending (ascending) chain $M\geq M_1 \geq M_2 \geq …$ ( $M_1 \leq M_2 \leq M_3 \leq …$ ) of submodules of $M$ , there exists an index $n \geq 1$ such that $M_n$ is isomorphic to $M_i$ for every $i\geq n$ . A ring $R$ is called right isoartinian (isonoetherian) if $R_R$ is an isoartinian (isonoetherian) $R$ -module. It is clear from the definitions that right artinian (noethrian) rings are right isoartinian (isonoetherian). We know that right artinian rings are right noetherian. Can we also say that right isoartinian rings are right isonoetherian?

References

[1] A. Facchini and Z. Nazemian, Modules with chain conditions up to isomorphism, Journal of Algebra, pp. 578–601, Vol. 453, 2016.

Krull-Schmidt theorem for Artin Algebras via evaluation functors

May 17 2022 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University.
Date: 17th of May, 2022, Tuesday, Time: 10.30 – 12.00.
Place: Dokuz Eylül University, Tınaztepe Campus, Faculty of Science, Department of Mathematics, room B206 (department seminar and meeting room).
Abstract: In this talk, we will show a process of passing from a given Artin Algebra to another one and find a correspondence between some of their module subcategories via the so called evaluation functors. This will help to prove the Krull-Schmidt theorem using the results in the previous talk.

Right minimal morphisms and projective covers over Artin Algebras

May 10 2022May 10 2022 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University.
Date: 10th of May, 2022, Tuesday, Time: 10.30 – 12.00.
Place: Dokuz Eylül University, Tınaztepe Campus, Faculty of Science, Department of Mathematics, room B206 (department seminar and meeting room).
Abstract: In this talk, we will introduce the basics of the representation theory of Artin Algebras. In particular, the notion of right minimal morphism will help us to prove the existence of a projective cover for any finitely generated module. We will also characterize the indecomposable projective modules over these Algebras.

Modules over Noetherian Hopf Algebras

April 19 2022April 19 2022 by Zübeyir Türkoğlu

Christian Lomp, University of Porto, Portugal.

Date: 21st of April, 2022, Thursday, Time: 10.30 – 12.00.

Place: Dokuz Eylül University, Tınaztepe Campus, Faculty of Science, Department of Mathematics, room B206 (department seminar and meeting room).

Abstract: In the first part of this talk I will give a short introduction to the theory of Hopf algebras. After surveying basic examples of Hopf algebras, their actions on rings as well as their basic properties I will report on my recent joint work with Can Hatipoglu on finiteness conditions on the injective hull of simple modules over Noetherian Hopf algebras of finite Gelfand-Kirillov dimension.

Nakayama Algebras

March 28 2022March 28 2022 by Zübeyir Türkoğlu

Engin Mermut, Dokuz Eylül University.

Date: 29th of March, 2022, Tuesday, Time: 10.00 – 12.00.

Place: Dokuz Eylül University, Tınaztepe Campus, Faculty of Science, Department of Mathematics, room B206 (department seminar and meeting room).

Abstract: Continuing our theme with seriality, we shall describe for which finite connected quivers Q, the bound quiver algebra KQ/I is a Nakayama algebra, that is, a serial algebra, where K is a field, KQ is the path algebra of the quiver Q and I is an admissible ideal of KQ. We shall summarize firstly what bound quiver algebras are. See Section I.10 in [1] and Chapter 10 in [2].

References [1] Skowroński, A. and Yamagata, K. Frobenius Algebras I, Basic Representation Theory. European Mathematical Society, 2012. [2] Drozd, Y. A. and Kirichenko, V. V. Finite Dimensional Algebras. Springer, 1994.