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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 6016 METHODS OF MODULE THEORY ELECTIVE 3 0 0 8

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 Mathematics

Course Objective

This course will cover the theory of special classes of modules and the theory of homological dimensions of modules and rings.

Learning Outcomes of the Course Unit

1   Will be able to use free, projective, injective and flat modules for the investigation of the properties of the category of modules. 2   Will be able to use Schanuel s lemma when computing projective and injective dimensions of modules. 3   Will be able to understand the homological classification of rings by their global dimension. 4   Will be able to use the uniform dimension of modules. 5   Will be able to understand the notions of essentiality, singularity and denseness of submodules.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Invariant basis number. The rank condition. 2 Dual basis lemma and invertible modules. Invertible fractional ideals. Hereditary artinian rings. Trace ideals. 3 Self-injective rings. Injectivitiy versus divisibilty. Essential extensions and injective hulls. 4 Injectives over right noetherian rings. Indecomposable injectives and uniform modules. Injectives over some artininan rings. 5 Simple injectives. Matlis Theory. Some computations of injective hulls. 6 Flatness tests. Finitely presented modules. Finitely generated flat modules. 7 Coherent modules and coherent rings. Faitfully flat modules. Pure exact sequences. 8 Midterm 9 Schanuel s lemma and projective dimensions. Change of rings. Injective dimensions. 10 Weak dimensions of rings. Global dimensions of semiprimary rings, local rings and commutative noetherian rings. 11 Complements and closed submodules. Essential closures. CS modules. 12 Finiteness conditions on rings. Quasi-injective modules. 13 Singular submodules and nonsingular rings. 14 Dense submodules and rational hulls.

Recomended or Required Reading

T. Y. Lam. Lectures on modules and rings. Vol. 189. Springer-Verlag, New York, 1999.

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving

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 *** 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 5 65 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) 200

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