Algebraic Structures I (MAY422)
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General
School 
School of Science 

Academic Unit 
Department of Mathematics 
Level of Studies 
Undergraduate 
Course Code 
MAY422 
Semester  4 
Course Title 
Algebraic Structures I 
Independent Teaching Activities 
Lectures (Weekly Teaching Hours: 5, Credits: 7.5) 
Course Type 
General Background 
Prerequisite Courses   
Language of Instruction and Examinations 
Greek, English 
Is the Course Offered to Erasmus Students 
Yes (in English) 
Course Website (URL)  See eCourse, the Learning Management System maintained by the University of Ioannina. 
Learning Outcomes
Learning outcomes 
The course aims to introduce the students to the study algebraic properties of sets which are equipped with one or more (binary) operations. Such mathematical objects are called algebraic structures. We will mainly deal with two types of algebraic structures:
We will formulate various theorems concerning the structure and basic properties of groups and rings emphasizing the concept of isomorphism of groups or rings. From the perspective of Algebra two algebraic structures which are isomorphic, they have exactly the same algebraic properties. As a direct consequence, results concerning an algebraic structure are valid in any isomorphic algebraic structure. In the course we present several examples illuminating various notions of symmetry. It should be noted that the notion of symmetry is the central theme which underlies the concept of group/ring.


General Competences 
The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the theory of algebraic structures, in particular of the general theory of Groups and Rings, which form an important part of modern algebra. The contact of the undergraduate student with the ideas and concepts of the theory of groups and 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
 Preliminaries: Sets, functions, equivalence relations, partitions, (binary) operations.
 Groups – Permutation groups.
 Cyclic groups – generators.
 Cosets with respect to a subgroup – Lagrange’s Theorem.
 Homomorphisms of groups – Quotient groups.
 Rings and fields  Integral domains.
 The theorems of Fermat and Euler.
 Polynomial rings – Homomorphisms of Rings.
 Quotient rings – Prime and maximal ideals.
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.
 
Teaching Methods 
 
Student Performance Evaluation 
Final written exam 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:
 