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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5003 MODULES AND RINGS - I 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 SALAHATTIN ÖZDEMIR

Offered to

Mathematics (English) Mathematics (English)

Course Objective

The aim of this course is to provide an introduction to the theory of rings and modules.

Learning Outcomes of the Course Unit

1   Will be able to understand the basics of ring theory and the fundamental properties of modules. 2   Will be able to use category theory that are needed to discuss module categories. 3   Will be able to use the chain conditions on rings and modules, and the properties of Noetherian and Artininan modules. 4   Will be able to use projective, injective andf flat modules for the investigations in module theory. 5   Will be able to use the powerful Wedderburn-Artin structure theorems for semisimple rings.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Classes, sets and functions. Ordinal and cardinal numbers. Commutative diagrams. 2 Rings and ideals. Factor rings. Ring homomorphisms. 3 Modules and module homomorphisms. Factor modules. 4 Direct products and direct sums. Free modules. 5 Rings with Invariant Basis Number. Tensor products of modules. 6 Categories and functors. Exact sequences of modules. 7 Hom and Tensor as functors. Equivalent categories and adjoint functors. 8 Problem solving. 9 Generating and cogenerating classes. Noetherian and artinian modules. 10 Modules over principal ideal domains. Free modules and finitely generated modules over a PID. 11 Injective and projective modules. Hereditary and semihereditary rings. 12 Flat modules. Coherent and regular rings. Quasi-injective and quasi-projective modules. 13 The Jacobson and the prime radical. Radicals and chain conditions. 14 Wedderburn-Artin theory. Primitive rings and density. Semisimple rings.

Recomended or Required Reading

[1] Paul E. Bland. Rings and their modules. Walter de Gruyter & Co., Berlin, 2011. [2] Friedrich Kasch. Modules and rings. Translated from the German by D.A.R. Wallace. Vol. 17. Academic Press, London-New York, 1982. [3] 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 1 MIDTERM EXAM 1 2 MTE 2 MIDTERM EXAM 2 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE MTE 1 * 0.30 + MTE 2 * 0.30 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE 1 * 0.30 + MTE 2 * 0.30 +RST * 0.40

Further Notes About Assessment Methods

None

Assessment Criteria

The passing grade will be evaluated as 30% two midterm exams and 40% final exam. Class participation is important.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

e-mail: salahattin.ozdemir@deu.edu.tr Office: B 351/1 Phone: (232) 301 86 08

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 14 3 42 Preparations before/after weekly lectures 14 4 56 Preparation for midterm exam 2 20 40 Preparation for final exam 1 30 30 Final 1 3 3 Midterm 2 2 4 TOTAL WORKLOAD (hours) 175

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 4 3 2 2 4 4 3 LO.2 4 4 2 2 2 3 3 3 LO.3 3 3 3 2 2 4 3 3 LO.4 3 3 2 2 2 4 3 2 LO.5 3 3 2 2 2 3 3 2