Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-40-11-M-4

Commutative Algebra (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-40-11-M-4 Commutative Algebra 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. in WiSe
Level [4] Bachelor (Specialization)
Language [EN] English
Module Manager
Lecturers
+ further Lecturers of the department Mathematics
Area of study [MAT-AGCA] Algebra, Geometry and Computer Algebra
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Notice

Without a proof of successful participation in the exercise classes, only 6 credit points will be awarded for the module.

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V+2U MAT-40-11-K-4
Commutative Algebra
P 84 h 186 h
U-Schein
- PL1 9.0 WiSe
  • About [MAT-40-11-K-4]: Title: "Commutative Algebra"; Presence-Time: 84 h; Self-Study: 186 h
  • About [MAT-40-11-K-4]: The study achievement [U-Schein] proof of successful participation in the exercise classes (ungraded) must be obtained.

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 84120 ("Commutative Algebra")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

  • Rings, modules, localization, Nakayama's lemma
  • Noetherian / Artinian rings and modules
  • Primary decomposition
  • Krull's Principal Ideal Theorem, Krull dimension
  • Integral ring extensions, Going-up, Going-down, normalization
  • Noether normalization, Hilbert's Nullstellensatz
  • Dedekind Domains, invertible ideals

Competencies / intended learning achievements

Upon completion of this module, students have studied and understand the language and methods of commutative algebra, which is necessary to continue studying in the area of algebraic geometry, computer algebra and number theory. They have recognized how taking a higher point of view, that is, the abstraction of the problem, makes it possible at once to treat and solve completely different questions simultaneously. They understand the proofs presented in the lecture and are able to reproduce and explain them.

By completing the given exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learned to transfer the methods to new problems, to analyze them and to develop solution strategies.

Literature

  • M.F. Atiyah, I.G. Macdonald: Introduction to commutative algebra,
  • H. Matsumura: Commutative Ring Theory,
  • H. Matsumura: Commutative Algebra,
  • D. Eisenbud: Commutative Algebra with a View towards Algebraic Geometry,
  • G.-M. Greuel, G. Pfister: A Singular Introduction to Commutative Algebra.

Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de)

References to Module / Module Number [MAT-40-11-M-4]

Course of Study Section Choice/Obligation
[INF-88.79-SG] M.Sc. Computer Science Formal Fundamentals [WP] Compulsory Elective
[MAT-88.105-SG] M.Sc. Mathematics Pure Mathematics [WP] Compulsory Elective
[MAT-88.706-SG] M.Sc. Mathematics International Pure Mathematics [WP] Compulsory Elective
Module-Pool Name
[MAT-GM-MPOOL-5] General Mathematics (Introductory Modules M.Sc.)