Meltem Güllüsaç, Dokuz Eylül University.

Date: 20th of December, 2017, Wednesday. Time: 09:30 – 12:00.

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

Abstract: After a review of prime modules, prime ideals and indecomposable injectives, we will prove Matlis’s Theorem. See Chapter 3 of the monograph [1].

References

[1] Lam, T. Y. Lectures on Modules and Rings. Springer, 1999.

# Year: 2017

## Mininjective Rings

Meltem Güllüsaç, Dokuz Eylül University.

Date: 22nd and 29th of November, 6th and 13th of December, 2017, Wednesday. Time: 09:30 – 12:00.

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

Abstract: We shall make an introduction to mininjective rings. See the second chapter of the book by Nicholson and Yousif.

References

[1] Nicholson, W.K. and Yousif, M.F. Quasi-Frobenius Rings. Cambridge University Press, 2003.

## Quasi-Frobenius Rings

Noyan Er, Dokuz Eylül University.

Date: 25th of October, 1st, 8th and 15th of November, 2017, Wednesday. Time: 09:30-10:30

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

Abstract: This is the first of a series of talks on QF-rings and related questions.

## On Almost Perfect Rings

Sinem Benli, İzmir Institute of Technology.

Date: 18th of October, 2017, Wednesday. Time: 09:30-10:30

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

Abstract: In this talk, we shall give a brief summary about the motivating ideas for commutative almost perfect domains. After that, we can mention the generalization of this class of rings to noncommutative setting considered by Facchini and Parolin.

References:

[1] Facchini, A. and Parolin, C. Rings Whose Proper Factors are Right Perfect. *Colloquium Mathematicae*, 122, 191-202, 2011.

[2] Benli, S. *Almost Perfect Rings*. M.Sc. Thesis, Dokuz Eylül University, The Graduate School of Natural and Applied Sciences. İzmir/TURKEY, 2015.

## Department Seminar – Başak Karpuz

Speaker: Assoc. Prof. Dr. Başak Karpuz

Title: First-Order Linear Delay Differential Equations: Oscillation and Stability

Place: Department of Mathematics Seminar Room (B206)

Time: 13.30

Date: 13.10.2017

## On Periodic Modules

Salahattin Özdemir, Dokuz Eylül University.

Date: 27th of September, 4th and 11th of October, 2017, Wednesday. Time: 09:30 – 12:00.

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

Abstract: We will prove the well-known result by Benson and Goodearl about flat modules over an arbitrary ring *R*: If a flat *R*-module *M* sits in a short exact sequence *0* → *M* → *P* → *M* → *0* with *P* projective, then *M* is projective. In other words, every flat periodic *R*-module *M* (of period 1) is projective. We will then talk about some of the recent generalizations of this result.

## Jacobson Density Theorem and Its Applications

Sinem Benli, İzmir Institute of Technology.

Date: 20th of September, 2017, Wednesday. Time: 09:30-12:00

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

Abstract: Firstly, we introduce the notion of primitive rings. After giving some examples and mentioning the properties of this class of rings, we shall prove the famous Jacobson Density Theorem which gives the structure of the primitive rings. Finally, we give a different proof of the fundamental Wedderburn’s structure theorem that characterizes the finite dimensional simple algebras.

References:

[1] Matej Brešar, *Introduction to Noncommutative Algebra*, Springer, 2014.

[2] Benson Farb & R. Keith Dennis, *Noncommutative Algebra*, Springer, 1991.

## Skolem-Noether Theorem and Double Centralizer Theorem

Meltem Güllüsaç and Hikmet Burak Özcan, Dokuz Eylül University.

Date: 9th, 16th and 23rd of August and 13th of September, 2017, Wednesday, Time: 09:30 – 12:00.

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

Abstract: After a review of tensor product of vector spaces and tensor product of algebras over a field, we shall prove two classical results for finite-dimensional central simple algebras over a field: the Skolem-Noether Theorem and the Double Centralizer Theorem. See Chapter 4 of the book [1] by Matej Brešar. All the algebras we mention are algebras over a field. Multiplication algebra of an algebra is a useful tool, see for example the article [2] by Brešar with some results for zero product determined algebras. The fact that every linear operator on a finite dimensional central simple algebra *A* belongs to the multiplication algebra of *A* is used to give the following particular case of the Skolem-Noether theorem: every automorphism of a finite dimensional central simple algebra *A* is inner. Indeed a somewhat more general result is proved in [2] for the linear operators *f* and *g* on *A* that satisfy* f*(*x*)*g*(*y*)=0 whenever *xy*=0. In the recent preprint [3] by Brešar et. al., they investigate the unital algebras *S* such that for every finite-dimensional central simple algebra *R*, every homomorphism from *R* to the tensor product of the algebras *R* and *S* extends to an inner automorphism of this tensor product algebra. They call such algebras Skolem-Noether algebras. By the classical Skolem-Noether Theorem, every finite-dimensional central simple algebra is a Skolem-Noether algebra and in this work, they show that various classical and important families of algebras like semilocal (in particular artinian and finite-dimensional) algebras, unique factorization domains, free algebras, etc. are Skolem-Noether algebras.

References

[1] Matej Brešar, *Introduction to Noncommutative Algebras*, Springer, 2014.

[2] Matej Brešar, *Multiplication algebra and maps determined by zero products*, Linear and Multilinear Algebra, 60:7, 763-768, 2012.

[3] Matej Brešar, Christoph Hanselka , Igor Klep and Jurij Volčič, *Skolem-Noether Algebras*, preprint, arXiv.org > math > arXiv:1706.08976, 2017.

## Elementary Proofs of the Wedderburn-Artin Theorem

Meltem Güllüsaç, Hikmet Burak Özcan and Sedef Taşkın, Dokuz Eylül University.

Date: 2nd of August, 2017, Wednesday, Time: 09:30 – 17:00.

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

Abstract: We shall present three proofs of the Wedderburn-Artin Theorem. In the first proof for non-unital rings, we follow Szele’s article [1], firstly proving the Density Theorem. In the second proof for unital rings, we follow Nicholson’s article [2]. Both of these proofs are for semiprime right or left artinian rings. In the third proof, we follow Section 3 Structure of Semisimple Rings from the book [3] by Lam. All proofs are elementary.

References

[1] Szele, T., Simple proof of the Wedderburn-Artin structure theorem, Acta Mathematica Hungarica, 5(1-2), 101-107, 1954.

[2] Nicholson, W. K., A Short Proof of the Wedderburn Artin Theorem, New Zealand Journal of Mathematics, 22, 83-86, 1993.

[3] Lam, T. Y. A First Course in Noncommutative Rings. 2nd edition. Springer, 2001.

## Simple Rings and Central Algebras

Meltem Güllüsaç, Dokuz Eylül University.

Date: 21st of June, 2017, Wednesday, Time: 09:30 – 12:00.

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

Abstract: We shall make an introduction to simple rings and central algebras. See the first chapter of the book by Matej Brešar.

References

[1] Matej Brešar, Introduction to Noncommutative Algebras, Springer, 2014.