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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5044 MODULES AND RINGS - 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 Mathematics

Course Objective

The aim of this course is to study further topics in ring and module theory, including an introduction to homological algebra.

Learning Outcomes of the Course Unit

1   Will be able to understand that every module has an injective envelope and that, for left perfect rings, every left module has a projective cover. 2   Will be able to understand the method of constructing quotients for suitable rings which plays a role in the study of commutative algebra. 3   Will be able to use the basic methods for graded rings and modules. 4   Will be able to use the characterization of the rings related with the reflexivity of modules like quasi-Frobenius rings. 5   Will be able to use projective and injective resolutions of modules to obtain the left and right derived functors of the Hom functor and tensor product functor.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Injective envelopes and projective covers. Semiperfect and perfect rings. 2 Quasi-injective envelopes and quasi-projective covers. 3 Rings of quotients and modules of quotients. 4 Goldie s theorem. The maximal ring of quotients. 5 Graded rings and modules. Graded direct products and sums. Graded tensor products and graded free modules. 6 Graded projective, graded injective and graded flat modules. 7 Graded modules with chain conditions. Graded Wedderburn-Artin theory. 8 Midterm 9 Reflexivity and Vector Spaces. Kasch rings and injective cogenerators. Quasi-Frobenius rings. 10 Chain and cochain complexes. Projective and injective resolutions. 11 Derived functors. Extension and torsion functors. 12 Projective and injective dimension. 13 Flat dimension. Dimension of polynomial rings. 14 Dimension of matrix rings. More on reflexive modules.

Recomended or Required Reading

[1] Paul E. Bland. Rings and their modules. Walter de Gruyter & Co., Berlin, 2011. [2] Joseph J. Rotman. An introduction to homological algebra. Second edition. Springer, 2009.

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 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 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