algebra

Modules with chain conditions up to isomorphism II

by

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

by

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

by

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

by

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

by

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

by

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.

Krull-Schmidt via the exchange property

by

 

Victor Blasco Jimenez, Dokuz Eylül University.
Date: 8th 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: We will define what is called the exchange property and study modules for which this property holds. Later we will see how it can be used to prove the Krull-Schmidt Theorem and related ones for various classes of modules.


Artinian Serial Rings

by

Noyan ER, Dokuz Eylül University.
Date: 1st 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: We will continue our discussion of serial rings giving basic information about Artinian serial rings and presenting a standard proof that they have finite representation type.