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Special Topics in Algebra (MAE723): Διαφορά μεταξύ των αναθεωρήσεων

Special Topics in Algebra (MAE723): Διαφορά μεταξύ των αναθεωρήσεων

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* [[Ειδικά Θέματα Άλγεβρας (MAE723)|Ελληνική Έκδοση]]
* [[Ειδικά Θέματα Άλγεβρας (MAE723)|Ελληνική Έκδοση]]
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=== General ===
=== General ===
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! Learning outcomes
! 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 modules and rings. 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 principal aim of the course is to introduce the students to the main ideas and methods of Commutative Algebra.
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! General Competences
! General Competences
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The contact of the undergraduate student with the ideas and concepts of the theory of modules 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.
The course promotes inductive and creative thinking and aims to provide the student with the theoretical background and skills of commutative rings.
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=== Syllabus ===
=== Syllabus ===
* Elementary Ring Theory.
* Polynomial Rings
* Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains.
* Hilbert's Basis Theorem
* Module Theory.
* Localization
* Modules over polynomial rings.
* Integral dependence
* Finitely generated and free modules.
* Hilbert Series
* Modules over Principal Ideal Domains.
* Dimension
* Decomposition Theorems.
* Groebner Bases
* Applications to Linear Algebra and Abelian groups.
* Hilbert's Nullstellensatz Theorem
 
=== Teaching and Learning Methods - Evaluation ===
=== Teaching and Learning Methods - Evaluation ===
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{| class="wikitable"
Γραμμή 78: Γραμμή 80:
! Delivery
! Delivery
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Classroom (face-to-face)
Teaching on the blackboard by the teacher.
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! Use of Information and Communications Technology
! Use of Information and Communications Technology
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Communication with the teacher by electronic means (i.e. e-mail).
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! Teaching Methods
! Teaching Methods
Γραμμή 92: Γραμμή 95:
| 39
| 39
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| Working independently
| Personal study
| 78
| 78
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| Exercises-Homeworks
| Solving exercises
| 33
| 33
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Γραμμή 106: Γραμμή 109:
Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.  
Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.  
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=== Attached Bibliography ===
=== Attached Bibliography ===


Τελευταία αναθεώρηση της 12:31, 15 Ιουνίου 2023

General

School

School of Science

Academic Unit

Department of Mathematics

Level of Studies

Undergraduate

Course Code

MAE723

Semester

7

Course Title

Special Topics in Algebra

Independent Teaching Activities

Lectures (Weekly Teaching Hours: 3, Credits: 6)

Course Type

Special Background

Prerequisite Courses -
Language of Instruction and Examinations

Greek

Is the Course Offered to Erasmus Students

Yes

Course Website (URL) 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 ideas and methods of Commutative Algebra.

General Competences

The course promotes inductive and creative thinking and aims to provide the student with the theoretical background and skills of commutative rings.

Syllabus

  • Polynomial Rings
  • Hilbert's Basis Theorem
  • Localization
  • Integral dependence
  • Hilbert Series
  • Dimension
  • Groebner Bases
  • Hilbert's Nullstellensatz Theorem

Teaching and Learning Methods - Evaluation

Delivery

Teaching on the blackboard by the teacher.

Use of Information and Communications Technology

Communication with the teacher by electronic means (i.e. e-mail).

Teaching Methods
Activity Semester Workload
Lectures (13X3) 39
Personal study 78
Solving exercises 33
Course total 150
Student Performance Evaluation

Final written exam in Greek (in case of Erasmus students in English) which includes 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:

  • J.Beachy, Introductory Lectures on Rings and Modules, LMS, Cambridge University Press, (1999).
  • D.Dummit, R.M.Foote, Abstract Algebra, 3 edition, Prentice Hall, (2003).
  • N.Jacobson, Basic Algebra I & II, W. H. Freeman and Company, (1985 & 1989).
  • S.Lang, Algebra, Graduate Texts in Mathematics, Springer (2002).
  • L.Rowen, Ring Theory, Academic Press, 2 edition (1991).
  • Μαλιάκας. Ταλέλλη, Πρότυπα πάνω από Περιοχές Κυρίων Ιδεωδών και Εφαρμογές, Εκδ. Σοφία (2009).
  • Α. Μπεληγιάννης, Μια Εισαγωγή στη Βασική Άλγεβρα, Εκδ. Κάλλιπος (2015).
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