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Bachelor	TR-NQF-HE: Level 6	QF-EHEA: First Cycle	EQF-LLL: Level 6

Course Introduction and Application Information

Course Code	Course Name	Semester	Theoretical	Practical	Credit	ECTS
MAT4052	Commutative Algebra	Spring	3	0	3	6

This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester.

Basic information

Language of instruction:	English
Type of course:	Non-Departmental Elective
Course Level:	Bachelor’s Degree (First Cycle)
Mode of Delivery:	Face to face
Course Coordinator :
Recommended Optional Program Components:	None
Course Objectives:	To provide the necessary background (both computational and theoretical) in commutative algebra to mathematics majors.

Learning Outcomes

The students who have succeeded in this course;
A student who finishes this course successfully will have learned basic concepts of commutative algebra.

Course Content

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.

Weekly Detailed Course Contents

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.

Sources

Course Notes / Textbooks:	Instructor's own lecture notes. Atiyah and MacDonald, "Introduction to Commutative Algebra"
References:

Evaluation System

Semester Requirements	Number of Activities	Level of Contribution
Quizzes	3	% 10
Midterms	2	% 40
Final	1	% 50
Total		% 100
PERCENTAGE OF SEMESTER WORK		% 50
PERCENTAGE OF FINAL WORK		% 50
Total		% 100

ECTS / Workload Table

Activities	Number of Activities	Duration (Hours)	Workload
Course Hours	14	3	42
Study Hours Out of Class	14	2	28
Quizzes	3	3	9
Midterms	2	10	20
Final	1	26	26
Total Workload			125

Contribution of Learning Outcomes to Programme Outcomes

No Effect	1 Lowest	2 Low	3 Average	4 High	5 Highest

	Program Outcomes	Level of Contribution
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