What is commutative algebra

Commutative algebra

4 + 2 SWS

Target group: Math students from the 5th semester.
Prerequisite: lecture computer algebra.

Table of Contents

Subject of the lecture

The central object of commutative algebra are modules.
As is well known, modules are a natural generalization of vector spaces:
the main difference is that the scalars do not consist of a body,
but come from a (mostly commutative, hence the title) ring.
Apparently small cause, big effect:
In contrast to the vector space, a module usually has no basis,
and we lose the concept of dimension so practical in linear algebra.
On the other hand, rings have a much richer structure than bodies, and so can
significantly more interesting phenomena occur with modules than with vector spaces.
We will take a closer look at some of them in this lecture.

Commutative algebra is mainly used in
of algebraic geometry, coding theory and combinatorics,
as well as in the study of linear differential equation systems, but also
in apparently distant mathematical disciplines
like approximation theory, optimization and statistics.

literature

  • M. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books 1999
  • D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer 1995
  • G.-M. Greuel, G. Pfister, A Singular Introduction to Commutative Algebra, Springer 2007
  • E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser 1985
  • T. Y. Lam, Lectures on Modules and Rings, Springer 1998
  • D. G. Northcott, Finite Free Resolutions, Cambridge University Press 2004
  • R. Y. Sharp, Steps in Commutative Algebra, Cambridge University Press 2000