COURSE UNIT TITLE

: COMUTATIVE RING THEORY - II

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5040 COMUTATIVE RING THEORY - II 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 introduce further methods in studying commutative rings and modules over them.

Learning Outcomes of the Course Unit

1   Will be able to use primary decomposition of ideals.
2   Will be able to understand the notion of flatness and related concepts.
3   Will be able to use the algorithms in computational commutative algebra that has been developed using Groebner basis.
4   Will be able to use the properties of graded rings, Cohen-Macaulay rings, regular local rings and Gorenstein rings.
5   Will be able to understand the notion of completion and the structure of complete local rings.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Associated primes and primary decomposition. Secondary representations of a module.
2 Flatness. Pure submodules. Flat families. Completions and hensel s lemma.
3 Filtrations and the Artin-Rees lemma.
4 Introduction to Dimension Theory. Systems of parameters.
5 Graded rings. The Hilbert function and the Samuel function. The principal ideal theorem.
6 Groebner basis.
7 Regular sequences.
8 The Koszul complex.
9 Depth, codimension and Cohen-Macaulay rings.
10 Regular local rings.
11 Free resolutions and Fitting invariants.
12 Gorenstein rings.
13 Derivations and differentials. Seperability. Higher derivations.
14 I-smoothness. The structure theorem for complete local rings.

Recomended or Required Reading

Textbook(s):
H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.

Supplementary Book(s):
[1] David Eisenbud, Commutative Algebra: with a View Toward Algebraic Geometry, Springer, 1999.
[2] Irving Kaplansky, Commutative Rings, The University of Chicago Press, 1974.
[3] J. P. Serre, Local algebra, Springer, 2000.
[4] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison Wesley, 1994.
[5] D. W. Sharpe and P. Vámos, Injective modules, Cambridge University Press, 1972.
[6] Rodney Y. Sharp, Steps in Commutative Algebra, 2nd edition, Cambridge University Press, 2004.

Planned Learning Activities and Teaching Methods

Lecture notes, presentation, problem solving, discussion.

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 MTE MIDTERM EXAM
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40


Further Notes About Assessment Methods

Homework
1 Midterm Exam
Final Exam

Assessment Criteria

%30 (Homework) + %30 (Midterm examination) +%40 (Final examination)

Language of Instruction

English

Course Policies and Rules

You can be successful in this course by studying from your textbooks and lecture notes on the topics to be covered every week, coming to class by solving the given problems, establishing the concepts by discussing the parts you do not understand with your questions, learning the methods, and actively participating in the course.

Contact Details for the Lecturer(s)

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

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 3 42
Preparations before/after weekly lectures 14 3 42
Preparation for midterm exam 1 15 15
Preparation for final exam 1 20 20
Preparing assignments 10 5 50
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 175

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.1433222433
LO.2433222433
LO.3433222433
LO.4433222433
LO.5433222433

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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