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

Τελευταία αναθεώρηση της 12:31, 15 Ιουνίου 2023

Ελληνική Έκδοση
Undergraduate Courses Outlines
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Department of Mathematics
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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).

@@ Γραμμή 1: / Γραμμή 1: @@
-[[Undergraduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
+* [[Ειδικά Θέματα Άλγεβρας (MAE723)|Ελληνική Έκδοση]]
+{{Course-UnderGraduate-Top-EN}}
+{{Menu-OnAllPages-EN}}
 === General ===
@@ Γραμμή 30: / Γραμμή 32: @@
 ! Independent Teaching Activities
 |
-Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
+Lectures (Weekly Teaching Hours: 3, Credits: 6)
 |-
 ! Course Type
@@ Γραμμή 56: / Γραμμή 58: @@
 ! Learning outcomes
 |
-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.
 |-
 ! General Competences
 |
-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.
 |}
 === 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 ===
 {| class="wikitable"
@@ Γραμμή 77: / Γραμμή 80: @@
 ! Delivery
 |
-Classroom (face-to-face)
+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
@@ Γραμμή 91: / Γραμμή 95: @@
 | 39
 |-
-| Working independently
+| Personal study
 | 78
 |-
-| Exercises-Homeworks
+| Solving exercises
 | 33
 |-
@@ Γραμμή 105: / Γραμμή 109: @@
 Final written exam in Greek (in case of Erasmus students in English) which includes 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:
+<!-- In order to edit the bibliography, visit the webpage -->
-* Μ. Μαλιάκας - Ο. Ταλέλλη: «Πρότυπα πάνω σε Περιοχές Κυρίων Ιδεωδών και Εφαρμογές»,  Εκδόσεις Σοφία.
+<!-- https://wiki.math.uoi.gr/index.php/%CE%A0%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF:MAE723-Biblio -->
-* Μ. Μαλιάκας: «Εισαγωγή στη Μεταθετική Άλγεβρα»,  Εκδόσεις Σοφία.
-* N. Jacobson: “Basic Algebra I”, Dover Publications (1985).
+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. Books and other resources, not provided by Eudoxus:
-* S. Lang: «Άλγεβρα», Εκδόσεις Πολιτεία (2010).
+{{MAE723-Biblio}}