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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5038 NONCOMMUTATIVE RINGS ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics (English) Mathematics (English)

Course Objective

The aim of this course is to give a general introduction to the theory of noncommutative rings.

Learning Outcomes of the Course Unit

1   Will be able to use the structure of semisimple rings and primitive rings. 2   Will be able to understand the proof of Hopkins-Levitzki Theorem, using the notion of Jacobson radical. 3   Will be able to realize the fundamental connections between ring theory and the representation theory of groups. 4   Will be able to use the properties of local and semilocal rings. 5   Will be able to understand the homological characterizations of perfect and semiperfect rings.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Wedderburn-Artin theory of semisimple rings. 2 The Jacobson radical. Jacobson radical under change of rings. 3 Group rings and the J-semisimplicity problem. 4 Modules over finite-dimensional algebras. 5 Representations of groups. 6 Linear groups. 7 The prime radical; prime and semiprime rings. Structure of primitive rings; the Density theorem. 8 Midterm 9 Division rings. Tensor products and maximal subfields. 10 Ordering and preordering in rings. Ordered division rings. 11 Local and semilocal rings. 12 The theory of idempotents. 13 Perfect and semiperfect rings. 14 Homological characterization of perfect and semiperfect rings.

Recomended or Required Reading

Textbook(s): [1] T. Y. Lam. A first course in noncommutative rings. Vol. 131. Springer-Verlag, New York, 2001. [2] F. W. Anderson and K. R. Fuller. Rings and categories of modules. Springer, New York, 1992.

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 FIN FINAL EXAM 3 FCG FINAL COURSE GRADE MTE * 0.50 + FIN * 0.50 4 RST RESIT 5 FCGR FINAL COURSE GRADE MTE * 0.50 + RST * 0.50 *** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

e-mail: engin.mermut@deu.edu.tr Office: (232) 301 85 82

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 3 39 Preparations before/after weekly lectures 13 3 39 Preparation for midterm exam 1 15 15 Preparation for final exam 1 25 25 Preparing assignments 10 5 50 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 LO.1 4 3 3 3 2 4 3 2 LO.2 4 3 3 3 2 4 3 2 LO.3 4 3 3 3 2 4 4 3 LO.4 4 3 3 3 2 4 3 3 LO.5 4 3 3 3 2 4 3 3