| MATHEMATICS (TURKISH, PHD) | |||||
| PhD | TR-NQF-HE: Level 8 | QF-EHEA: Third Cycle | EQF-LLL: Level 8 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT4052 | Commutative Algebra | Fall | 3 | 0 | 3 | 6 |
| The course opens with the approval of the Department at the beginning of each semester |
| Language of instruction: | En |
| Type of course: | Departmental Elective |
| Course Level: | |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Dr. Öğr. Üyesi TUĞCAN DEMİR |
| Course Objectives: | To provide the necessary background (both computational and theoretical) in commutative algebra to mathematics majors. |
|
The students who have succeeded in this course; A student who finishes this course successfully will have learned basic concepts of commutative algebra. |
| Abelian groups, rings and fields. Vector spaces and linear transformations. Bases and matrix representations of linear transformations. Polynomial rings. Ideals, prime and maximal ideals. Quotients of polynomial rings. Modules over polynomial rings. Prime and primary ideals. Factorization of ideals in the monoid of ideals. Localizations of ideals. Zero-divisors, integral domains and rings of fractions. Unique factorization domains and Euclidean domains. Radical of an ideal. Nilradical and Jacobson radical of a ring. Operations in the lattice of ideals. Classical Euclidean division algorithm in polynomial algebras. Monomial orderings and division algorithms. Fundamental Theorem of Algebra. Finite generation of ideals in polynomial algebras. Gröbner basis and Buchberger algorithm. Examples and calculations. Gröbner bazları ve Buchberger algoritması. Örnekler ve hesaplamalar. Gröbner basis and Buchberger algorithm. Examples and calculations. Morphisms between modules. Kernels and images of morphisms. Submodules and quotient modules. Ideals of annihilators. Internal and external sums of modules. Tensor products of modules. Submodule and ideal chains. Artinian and Noetherian rings and modules. |
| Week | Subject | Related Preparation | |
| 1) | Abelian groups, rings and fields. | ||
| 2) | Vector spaces and linear transformations. Bases and matrix representations of linear transformations. | ||
| 3) | Polynomial rings. Ideals, prime and maximal ideals. Quotients of polynomial rings. Modules over polynomial rings. | ||
| 4) | Prime and primary ideals. Factorization of ideals in the monoid of ideals. Localizations of ideals. | ||
| 5) | Zero-divisors, integral domains and rings of fractions. Unique factorization domains and Eucledian domains. | ||
| 6) | Radical of an ideal. Nilradical and Jacobson radical of a ring. Operations in the lattice of ideals. | ||
| 7) | A review of covered subjects and the first exam. | ||
| 8) | Classical Euclidean division algorithm in polynomial algebras. Monomial orderings and division algorithms. | ||
| 9) | Fundamental Theorem of Algebra. Finite generation of ideals in polynomial algebras. | ||
| 10) | Gröbner basis and Buchberger algorithm. Examples and calculations. | ||
| 11) | Gröbner basis and Buchberger algorithm. Examples and calculations. | ||
| 12) | A review of covered subjects and the second exam. | ||
| 13) | Morphisms between modules. Kernels and images of morphisms. Submodules and quotient modules. Ideals of annihilators. Examples. | ||
| 14) | Internal and external sums of modules. Tensor products of modules. Submodule and ideal chains. Artinian and Noetherian rings and modules. | ||
| Course Notes: | Instructor's own lecture notes. Atiyah and MacDonald, "Introduction to Commutative Algebra" |
| References: |
| Semester Requirements | Number of Activities | Level of Contribution |
| Attendance | 0 | % 0 |
| Laboratory | 0 | % 0 |
| Application | 0 | % 0 |
| Field Work | 0 | % 0 |
| Special Course Internship (Work Placement) | 0 | % 0 |
| Quizzes | 3 | % 10 |
| Homework Assignments | 0 | % 0 |
| Presentation | 0 | % 0 |
| Project | 0 | % 0 |
| Seminar | 0 | % 0 |
| Midterms | 2 | % 40 |
| Preliminary Jury | % 0 | |
| Final | 1 | % 50 |
| Paper Submission | % 0 | |
| Jury | % 0 | |
| Bütünleme | % 0 | |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 50 | |
| PERCENTAGE OF FINAL WORK | % 50 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Laboratory | 0 | 0 | 0 |
| Application | 0 | 0 | 0 |
| Special Course Internship (Work Placement) | 0 | 0 | 0 |
| Field Work | 0 | 0 | 0 |
| Study Hours Out of Class | 14 | 2 | 28 |
| Presentations / Seminar | 0 | 0 | 0 |
| Project | 0 | 0 | 0 |
| Homework Assignments | 0 | 0 | 0 |
| Quizzes | 3 | 3 | 9 |
| Preliminary Jury | 0 | ||
| Midterms | 2 | 10 | 20 |
| Paper Submission | 0 | ||
| Jury | 0 | ||
| Final | 1 | 26 | 26 |
| Total Workload | 125 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution |