Zoek
English
  Studiegidsen 2011-2012
Radboud UniversiteitStudiegidsenFaculteit der Natuurwetenschappen, Wiskunde en Informatica > Master Mathematics

Commutative Algebra 

(Vakcode)
Course ID
NWI-WM026B
(Studiepunten)
Credits
8
(Periode)
Scheduled
eerste semester
Teaching methods (Werkvormen)
  • 28 hrs lecture
  • 28 hrs problem session
Objectives (Leerdoelen)
  • The student understands the difference in structure of the main objects in Commutative Algebra: rings, ideals, modules and algebras
  • The student can distinguish between different objects of the same type, like prime, maximal and nilpotent ideals, and Noetherian modules
  • The student can apply standard operations to a specific object, like localization, taking quotients, and tensoring
  • The student can apply results from Commutative Algebra to various problems encountered during this course
Contents (Beschrijving)
What is commutative algebra about? To make this clear let's start with a k-vector space V, where k is a field. So V is a set equipped with an addition, which makes V into an abelian group, and a scalar multiplication with scalars from k. Furthermore the classical distibutive laws hold. If we replace k by an arbitrary commutative ring R we get a so-called R-module. This notion generalises most of the notions one meets during a Bachelor's study Mathematics. For example it will turn out that a Z-module is the same as an abelian group, a k[x]-module is the same as a k-vector space together with a linear transformation and an ideal I in a ring R is an example of a so-called R-submodule of R. Also the quotient ring R/I is an R-module etc. The theory of R-modules is much more complicated than the theory of vector spaces; many problems are still unsolved. The general philosophy is that the 'nicer' the ring R is, the more we know about its R-modules. The language of modules is an indispensable tool in nowadays Mathematics. In this course we discuss the most fundamental concepts and results of modern commutative algebra. Many of the notions introduced in this course will also be used in various other courses.
If you are planning to specialize in algebraic geometry, algebraic topology, number theory, computer algebra or polynomial mappings, this course is a must.
Examination (Tentaminering)
The student has to make a series of exercises.
Literature (Literatuur)
A handout will be provided.
Recommendation: the excellent book 'Introduction to commutative algebra' by M.F. Atiyah and I.G. MacDonald.