Engin Mermut

Skolem-Noether Theorem and Double Centralizer Theorem

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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

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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

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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.

Finite Dimensional Division Algebras

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Hikmet Burak Özcan, Dokuz Eylül University.
Date: 14th 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 finite dimensional algebras over a field starting from finite dimensional division algebras. See the first chapters of the  book by Matej Brešar.
References
[1] Matej Brešar, Introduction to Noncommutative Algebras, Springer, 2014.

Leavitt Path Algebras

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Tuğba Güroğlu, Celal Bayar University.
Date: 24th and 31st of May, and 7th of June, 2017, Wednesday, Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: Let E denote a directed graph and K be a field. The Leavitt path algebra of E with coefficient in K, denoted by L_K(E), were introduced by G. Abrams and G. Aranda Pino in 2005 as a generalization of Leavitt algebras and then extended to arbitrary graphs in 2008. In this talk, we talk about Leavitt path algebras and determine Leavitt path algebras of some graphs. Then using graph-theoretic properties, we mention the some ring-theoretic properties of Leavitt path algebras.
References
[1] Abrams, G. and Aranda Pino, G., The Leavitt path algebra of a graph, J. Algebra, 293(2), 319-334, 2005.
[2] Abrams, G. and Aranda Pino, G., The Leavitt path algebras of arbitrary graphs, Houston J. Math., 34(2), 423-442, 2008.

Nonstandard Analysis and Primality Test

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Haydar Göral, Koç University.
Date: 17th of May, 2017, Wednesday, Time: 11:00 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In this talk, we first define the height function and the Mahler measure from Diophantine geometry on the field of algebraic numbers. The height function measures the arithmetic complexity of an algebraic number and it has some nice properties. Then, we explain nonstandard analysis which we apply to find certain height bounds. Nonstandard analysis was originated in the 1960’s by the work of A. Robinson, which arose as a rigorous and exhaustive way of studying infinitesimal calculus. Combining the properties of the height function with ideas from model theory and nonstandard analysis, we explain how to obtain some uniform height bounds in the polynomial ring over the field of algebraic numbers. This enables us to test the primality of an ideal.

Rings of Quotients

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M. Pınar Eroğlu, Dokuz Eylül University.
Date: 8th and 22nd of February, 8th of March, 26th of April, 2017, Wednesday, Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: See the monograph by Beidar and Martindale and Mikhalev (Rings with generalized identities)