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.
[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, > 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.
[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.