# Ring Theory (MAE725)

### General

School School of Science Department of Mathematics Undergraduate MAE725 7 Ring Theory Lectures (Weekly Teaching Hours: 3, Credits: 6) Special Background - Greek, English Yes See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes The principal aim of the course is to introduce the students to the main tools and methods of the theory of non-commutative rings, where by non-commutative ring is meant an associative ring with unit, which is not necessarily commutative. The main objective of the course is to present the basic theory of rings and the ideas which lead to the proof of: (a) the fundamental theorem of Wedderburn-Artin concerning the structure of semisimple rings and, (b) the fundamental density theorem of Jacobson concerning the structure of primitive rings. A key element in the study of a ring is the interaction and interplay between ring-theoretical properties of the ring and the structure of its (left or right) ideals or modules. In the course a variety of examples and constructions will be analyzed and various applications of ring theory to other areas of mathematics (in particular of algebra) will be explored. At the end of the course we expect the student to have understood the definitions and basic theorems which are discussed in the course, to have understood how they are applied in discrete examples, to be able to apply the material in order to extract new elementary conclusions, and finally to perform some (no so obvious) calculations. The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the Theory of Rings, which is an important part of modern algebra. The contact of the undergraduate student with the ideas and concepts of the theory of rings, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.

### Syllabus

Rings - Homomorphisms - Ideals - Quotient Rings - Modules - Rings arising from various constructions - Algebras - Group algebras - Modules over group algebras - Module homomorphisms - The bicommutator - Simple faithful modules and primitive rings - Artin rings - Simple finite dimensional algebras over algebraically closed fields - Artinian modules - Noetherian rings and modules - Jacobson radical.

### Teaching and Learning Methods - Evaluation

Delivery

Classroom (face to face)

Use of Information and Communications Technology
• Teaching Material: Teaching material in electronic form available at the home page of the course.
• Communication with the students:
1. Office hours for the students (questions and problem solving).
2. Email correspondence
3. Weekly updates of the homepage of the course.
Teaching Methods
Lectures (13x3) 39
Working independently 78
Exercises - Homeworks 33
Course total 150
Student Performance Evaluation

Combination of: Weekly homework, presentations in the class by the students, written work, and, at the end of the semester, written final exams in Greek (in case of Erasmus students, in English) which includes analysis of theoretical topics and resolving application problems.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Nathan Jacobson: "Basic Algebra I & II", W. H. Freeman and Company,  (1985 & 1989).
• I.N. Herstein: "Non-commutative Rings", AMS, Carus Mathematical Monographs 85, (1971).
• Luis Rowen: "Ring Theory (student edition)", Academic Press, Second Edition, (1991).
• T.Y. Lam: "A First Course in Noncommutative Rings", GTM 131, Springer, (2001).
• P. M. Cohn: "Introduction to Ring Theory", Springer (2000).
• Y. Drozd and V. Kirichenko: "Finite Dimensional Algebras", Springer (1994).