Προβολή πηγαίου κώδικα για τη σελίδα Ring Theory (MAE725)

[[Undergraduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]

=== General ===
{| class="wikitable"
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! School
|
School of Science
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! Academic Unit
|
Department of Mathematics
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! Level of Studies
|
Undergraduate
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! Course Code
|
MAE725
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! Semester
|
7
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! Course Title
|
Ring Theory
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! Independent Teaching Activities
|
Lectures (Weekly Teaching Hours: 3, Credits: 6)
|-
! Course Type
|
Special Background
|-
! Prerequisite Courses
| -
|-
! Language of Instruction and Examinations
|
Greek, English
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! Is the Course Offered to Erasmus Students
|
Yes
|-
! Course Website (URL)
| See [https://ecourse.uoi.gr/ eCourse], the Learning Management System maintained by the University of Ioannina.
|}

=== Learning Outcomes ===
{| class="wikitable"
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! Learning outcomes
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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.
<br/>
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.
<br/>
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. 
|-
! General Competences
|
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 ===
{| class="wikitable"
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! Delivery
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Classroom (face to face)
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! Use of Information and Communications Technology
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* Teaching Material: Teaching material in electronic form available at the home page of the course.
* Communication with the students:
# Office hours for the students (questions and problem solving). 
# Email correspondence
# Weekly updates of the homepage of the course. 
|-
! Teaching Methods
|
{| class="wikitable"
! Activity
! Semester Workload
|-
| 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 [https://service.eudoxus.gr/public/departments#20 Eudoxus site] or the [https://cloud.math.uoi.gr/index.php/s/62t8WPCwEXJK7oL local repository] of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Additionally:
* 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).