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Information
| Unit | INSTITUTE OF NATURAL AND APPLIED SCIENCES |
| MATHEMATICS (PhD) | |
| Code | MT014 |
| Name | Commutative Algebra |
| Term | 2018-2019 Academic Year |
| Term | Spring |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 6 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Belirsiz |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. YILMAZ DURĞUN |
| Course Instructor |
The current term course schedule has not been prepared yet.
|
Course Goal / Objective
In this course students will learn about commutative rings, subrings and ideals. Prime ideals and maxiaml ideals. Nilradical, Jacobson radical. Properties of ideals. Modules over commutative rings. Properties of Submodules. Direct products and direct sums. Finitely generated modules. Exact sequences. Tensor product. Exactness of Tensor product. Quotient rings.
Course Content
Commutative rings, subrings and ideals. Prime ideals and maxiaml ideals. Nilradical, Jacobson radical. Properties of ideals. Modules over commutative rings. Properties of Submodules. Direct products and direct sums. Finitely generated modules. Exact sequences. Tensor product. Exactness of Tensor product. Quotient rings.
Course Precondition
Resources
Notes
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | know the definition of commutative rings, local rings, prime and maximal ideals, and modules over commutative rings; |
| LO02 | are familiar with the notions of noetherian and artinian rings and modules; |
| LO03 | know how to localize rings and modules, and are familiar with important applications of localization; |
| LO04 | know the Hilbert basis theorem and the Hilbert Nullstellensatz; |
| LO05 | are familiar with the concepts of support and associated primes; |
| LO06 | know the definition of an exact sequence of modules, and you also know important properties and applications of exact sequences; |
| LO07 | know the concept of direct limit and you can compute this limit in some non-trivial examples; |
| LO08 | know how to define tensor products of modules and are familiar with the concept of flatness; |
| LO09 | know the basic results in the dimension theory for local rings; |
| LO10 | know how to complete a ring in an ideal. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Commutative rings, subrings and ideals. | Review of the relevant pages from lecture books | |
| 2 | Prime ideals and maxiaml ideals. Nilradical, Jacobson radical. | Review of the relevant pages from lecture books | |
| 3 | Properties of ideals. | Review of the relevant pages from lecture books | |
| 4 | Modules over commutative rings | Review of the relevant pages from lecture books | |
| 5 | Properties of Submodules. Direct products and direct sums. Finitely generated modules. | Review of the relevant pages from lecture books | |
| 6 | Noetherian rings and noetherian modules | Review of the relevant pages from lecture books | |
| 7 | Artinian rings and artinian modules | Review of the relevant pages from lecture books | |
| 8 | Mid-Term Exam | Review of the relevant pages from lecture books | |
| 9 | Modules over principal ideal domains | Review of the relevant pages from lecture books | |
| 10 | Canonical forms for square matrices | Review of the relevant pages from lecture books | |
| 11 | Some applications to field theory | Review of the relevant pages from lecture books | |
| 12 | Integral dependence on subrings | Review of the relevant pages from lecture books | |
| 13 | Dimension theory | Review of the relevant pages from lecture books | |
| 14 | Affine Algebras over fields | Review of the relevant pages from lecture books | |
| 15 | Cohen-Macaulay Rings | Review of the relevant pages from lecture books | |
| 16 | Term Exams | Review of the relevant pages from lecture books | |
| 17 | Term Exams | Review of the relevant pages from lecture books |