 Rings, modules, localization, Nakayama's lemma
 Noetherian / Artinian rings and modules
 Primary decomposition
 Krull's Principal Ideal Theorem, Krull dimension
 Integral ring extensions, Goingup, Goingdown, normalization
 Noether normalization, Hilbert's Nullstellensatz
 Dedekind Domains, invertible ideals
Module MAT4011M4
Commutative Algebra (M, 9.0 LP)
Module Identification
Module Number  Module Name  CP (Effort) 

MAT4011M4  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  [MATAGCA] Algebra, Geometry and Computer Algebra 
Reference course of study  [MAT88.105SG] M.Sc. Mathematics 
LivecycleState  [NORM] Active 
Courses
Type/SWS  Course Number  Title  Choice in ModulePart  PresenceTime / SelfStudy  SL  SL is required for exa.  PL  CP  Sem.  

4V+2U  MAT4011K4  Commutative Algebra
 P  84 h  186 h 
USchein
   PL1  9.0  WiSe 
 About [MAT4011K4]: Title: "Commutative Algebra"; PresenceTime: 84 h; SelfStudy: 186 h
 About [MAT4011K4]: The study achievement "[USchein] proof of successful participation in the exercise classes (ungraded)" must be obtained.
Examination achievement PL1
 Form of examination: oral examination (2030 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
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.unikl.de)
Requirements for attendance of the module (informal)
Modules:
Courses
 [MAT1211K2] Algebraic Structures (2V+2U, 5.5 LP)
 [MAT1222K3] Introduction to Algebra (2V+1U, 4.5 LP)
Requirements for attendance of the module (formal)
NoneReferences to Module / Module Number [MAT4011M4]
Course of Study  Section  Choice/Obligation 

[INF88.79SG] M.Sc. Computer Science  [Core Modules (non specialised)] Formal Fundamentals  [WP] Compulsory Elective 
[MAT88.105SG] M.Sc. Mathematics  [Core Modules (non specialised)] Pure Mathematics  [WP] Compulsory Elective 
[MAT88.706SG] M.Sc. Mathematics International  [Core Modules (non specialised)] Pure Mathematics  [WP] Compulsory Elective 
ModulePool  Name  
[MATGMMPOOL5]  General Mathematics (Introductory Modules M.Sc.) 
Notice