Sinem Odabaşı, Institute of Physics and Mathematics, Science Faculty, The Universidad Austral de Chile (UACh).

Date: 23rd of July, 2019, Tuesday. Time: 11:15 – 12:30.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: The ‘Phantom phenomenon’ has been sucessufelly carried into abelian setting firstly in [Her07], later in [FGHT13]. In this talk, we claim to introduce ‘ghost phenomenon’ in abelian setting which is also compatible with the existent ones in certain tringulated categories as mentioned above. Besides, we observe that the problem of being zero a finite power of ghost ideal in these triangulated categories is strongly related to being a certain type of object ideals and cotorsion pairs. Using this observation and certain techniques/results on cotorsion pairs, now we are able to ensure that under mild conditions the ideal ‘Ghost’ in an abelian category is always turned out to be ‘zero’ in some infinite power. We then apply this formalism to the ideal Ghost of chain morphisms which induce zero in homology in the category of chain complexes of left *R*-modules.

This is a joint work-in-progress with Sergio Estrada, X.H. Fu and Ivo Herzog, which has been supported by the grant CONICYT/FONDECYT/Iniciaci\’on/11170394.

References:

[FGHT13] Fu, X. H., Guil Asensio, P. A, Herzog, I. & Torecillas, B. (2013). Ideal approximation theory. Adv. Math. 244, 750-790.

[Her07] Herzog, I. (2007). The phantom cover of a module. Adv. Math. 215, 220–249.

# Engin Mermut

## On Isoartinian and Isonoetherian Modules – 2

Hakan Şanal, Dokuz Eylül University.

Date: 29th of May, 2019, Wednesday. Time: 14:30 – 16:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: We will continue the seminar with some examples of comparing right isoartinian (isonoetherian) rings and right artinian (noetherian) rings. Then, we deal with the endomorphism ring of an isosimple module.

References

[1] A. Facchini and Z. Nazemian, Modules with chain conditions up to isomorphism. J. Algebra 453 (2016): 578–601.

[2] A. Facchini and Z. Nazemian, Artinian dimension and isoradical of modules. J. Algebra 484 (2017): 66–87.

## On Isoartinian and Isonoetherian Modules

Hakan Şanal, Dokuz Eylül University.

Date: 22nd of May, 2019, Wednesday. Time: 14:30 – 16:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In [1, 2], Facchini and Nazemian generalize the idea of Artinian and Noetherian modules by considering the chain conditions up to isomorphism. They call a module *M* isoartinan (resp. isonoetherian) if, for every descending (resp. ascending) chain *M ≥ M*_{1 }*≥ M*_{2}* ≥ · · · *(resp. *M*_{1}* ≤ M*_{2}* ≤ M*_{3}* ≤ · · · *) of submodules of *M* , there exists an index *n ≥ 1* s.t. *M*_{n}* **≅**M*_{i} for every *i ≥ n*. Similarly, *M* is called isosimple if *M* is non-zero and every non-zero submodule of *M* is isomorphic to *M*. In this seminar, we will give some properties of these three classes of modules.

References

[1] A. Facchini and Z. Nazemian, Modules with chain conditions up to isomorphism. J. Algebra 453 (2016): 578–601.

[2] A. Facchini and Z. Nazemian, Artinian dimension and isoradical of modules. J. Algebra 484 (2017): 66–87.

## Conjugate Fields and Primitive Element Theorem

Hikmet Burak Özcan, İzmir Institute of Technology.

Date: 15th of May, 2019, Wednesday. Time: 14:30 – 16:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this talk, first we will recall what we did in the last seminar. Then we will mention conjugate fields and we will give the proof of the primitive element theorem.

## Continuation of Algebraic Number Theory

Sedef Taşkın, Dokuz Eylül University.

Date: 8th of May, 2019, Wednesday. Time: 14:30 – 16:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: In this talk, first we will recall what we did in the last seminar about integral elements. Then we will mention integrally closed rings and give some examples. Finally we will introduce algebraic elements and algebraic extensions.

## The Last Call For Algebraic Number Theory

Sedef Taşkın, Dokuz Eylül University.

Date: 3rd of April, 2019, Wednesday. Time: 14:30 – 16:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: This talk will be a continuation of the series of talks about algebraic number theory. First we start with integral elements and mention some properties. Then we introduce algebraic elements and algebraic extensions.

## Counting from the Bottom to the Top-2

Noyan Er, Dokuz Eylül University.

Date: 27th of March, 2019, Wednesday. Time: 10:00-12:00

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will finish up what was started last week. Counting things, however, will continue in the upcoming seminars.

## The Second Invitation to Algebraic Number Theory

Hikmet Burak Özcan, Dokuz Eylül University.

Date: 20th of March, 2019, Wednesday. Time: 10:00 – 12:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: This will be the second talk of the series devoted to algebraic number theory. First, we will briefly recall what we did in the first talk. Then we will state the elegant theorem, proved by Chevalley, which concerns about diophantine equations over a finite field.

## Counting from the Bottom to the Top

Noyan Er, Dokuz Eylül University.

Date: 13th of March, 2019, Wednesday. Time: 09:30-10:30

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: After finishing up a leftover result from my last talk, we will discuss, as a natural follow-up, a characterization of Artinian principal ideal rings due to Eisenbud and Griffith the proof of which involves counting vertically.

## An Invitation to Algebraic Number Theory

Hikmet Burak Özcan, Dokuz Eylül University.

Date: 6th of March, 2019, Wednesday. Time: 09:30 – 12:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

Abstract: This will be the first of a series of talks in algebraic number theory. During the series of talks, we will cover several chapters of the book “Algebraic Theory of Numbers” by Pierre Samuel. In the first talk, we will mention some theorems and their corollaries from Chapter 1.

References

[1] Pierre Samuel. *Algebraic Theory of Numbers*, Hermann, 1970.