Frobenius Rings, Quasi-Frobenius Rings
and their Applications

Frobenius algebras and their generalization, Quasi-Frobenius rings, is one of the foundation topics of Ring Theory. They were originally studied as an important tool of the representation theory of finite groups. Two important papers by Nakayama published in the early 40's, showed how these abstract notions allowed to simplify long and tedious computations in representation theory.

Recently, it has been seen that Frobenius algebras play an important role in a variety of topics, ranging from the algebraic treatment and axiomatic foundation of Topological Quantum Field Theory to the development of Coding Theory.

In the course we will present the classical results on Quasi-Frobenius rings, combined with some excursion with their recent applications. Special emphasis will be given to the connections with Coding Theory and, in particular, to the recent results by J. W. Honold characterizing finite Frobenius rings.

This course is intended to serve as a bridge between the coding theory oriented courses and the ring theoretical ones. Examples will be emphasized, and problem sessions will be also a crucial part of the course.

The main goal will be not only to present the subject, but also to encourage students to manipulate the ideas by themselves. To see how the theoretical concepts apply in particular examples and are helpful to understand them better.