Program Overview

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Commutative Algebra

Undergraduate Course

The course develops the theory of commutative rings, which are of fundamental significance in describing geometric and number theoretic ideas algebraically.

Objectives and Content

The course covers the following topics:

• Ideals in commutative rings
• Chain conditions for ideals
• Localization of commutative rings
• Modules over commutative rings
• Numerical invariants of commutative rings and modules
• Tensor products and exact sequences of modules
• Noetherian rings and Hilbert basis theorem
• Nullstellensatz
• Noether normalization
• Primary decomposition of ideals
• Gröbner bases
• Hilbert series and Hilbert polynomials
• Dimension theory for local rings

Learning Outcomes

On completion of the course, the student should have the following learning outcomes defined in terms of knowledge, skills, and general competence:

Knowledge

The student:

• Knows basic definitions concerning elements in rings, classes of rings, and ideals in commutative rings
• Knows constructions like tensor product and localization, and the basic theory for this
• Knows basic theory for noetherian rings and Hilbert basis theorem
• Knows basic theory for integral dependence, and the Noether normalization lemma
• Has insight into the correspondence between ideals in polynomial rings, and the corresponding geometric objects: affine varieties
• Knows basic theory for support and associated prime ideals of modules, and knows primary decomposition of ideals in noetherian rings
• Knows the theory of Gröbner bases and Buchbergers algorithm
• Knows the theory of Hilbert series and Hilbert polynomials
• Knows dimension theory of local rings

Skills

The student:

• Can use algebraic tools which are important for many problems and much theory development in algebra, algebraic geometry, number theory, and topology
• Has solid experience and training in reasoning with abstract and general algebraic structures

General Competence

The student:

• Has insight into the most important algebraic theory which is used in other parts of mathematics
• Has insight into the mathematics that is used in computer algebra
• Sees the usefulness of abstract theory development so that different parts of mathematics, like number theory and algebraic geometry, can be described in the same framework

Course Details

• ECTS credits: 10
• Teaching semesters: Autumn
• Course code: MAT224
• Number of semesters: 1
• Teaching language: English
• Level of study: Undergraduate
• Semester of instruction: Autumn

Assessment

• Forms of assessment: Oral examination
• Grading scale: A to F, where Grade A is the highest passing grade and Grade F is a fail
• Assessment semester: Exam only in autumn semester
• Compulsory assignments and attendance: Exercises

Previous Knowledge

• Required previous knowledge: None
• Recommended previous knowledge: MAT220 Algebra
• Credit reduction due to course overlap: M221: 10 ECTS