# Commutative Algebra

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This wikibook shall give an introduction to commutative algebra, that is, the study of commutative algebraic objects. We'll learn about modules, algebrae, commutative rings. We'll not cover vector spaces in depth, since they are usually treated within linear algebra courses, and we'll also not cover the basics on groups, rings and fields, which are covered by the wikibook Abstract Algebra. We'll also not cover topological vector spaces, since their study does not belong purely to the realm of algebra, but also to topology.

## Contents

## Table of Contents[edit]

### Lattices[edit]

- Definitions and examples, Hasse diagrams
- An equivalent definition, the algebraic point of view
- Ordered vector spaces, possibly with more structure

### Modules[edit]

- The lattice of submodules, Noether isomorphism theorems
- Generators, chain conditions
- Determinants, Cayley–Hamilton and weak Nakayama
- Products and coproducts, the tensor product
- Localisation of modules
- Sequences of modules
- Torsion-free, flat, projective and free modules
- Normal and composition series
- Algebras

### Commutative rings[edit]

- Basics on prime and maximal ideals and local rings
- Radicals, strong Nakayama
- Spectrum with Zariski topology
- Jacobson rings and Jacobson spaces
- Noetherian rings and spaces
- Primary decomposition or Lasker–Noether theory
- Artinian rings
- Intersection and prime chains or Krull theory
- Valuations, (discrete) valuation rings
- Integral dependence

For further aspects of the theory, see the wikibook More Commutative Algebra.

## Sources[edit]

### Books[edit]

- Mac Lane, Saunders (1978).
*Categories for the Working Mathematician*. New York: Springer. - Zariski, Oscar; Samuel, Pierre (1958).
*Commutative Algebra*. Princeton, New Jersey: D. Van Nostrand Company. - Lang, Serge (2005).
*Algebra*. New York: Springer. - Jacobson, Nathan (1989).
*Basic Algebra II*. New York: Freeman & Co.. - Atiyah, Michael; Macdonald, Ian (1969).
*Introduction to Commutative Algebra*. Reading, Massachusetts: Addison-Wesley.

### Lecture notes[edit]

- Milne, James (2014),
*Algebraic Number Theory*, http://www.jmilne.org/math/CourseNotes/ANT.pdf - Höchster, Melvin (2010),
*Noether normalization and Hilbert's Nullstellensatz*, http://www.math.lsa.umich.edu/~hochster/615W10/supNoeth.pdf

### Papers[edit]

- Azarang, Alborz (2015),
*A one-line undergraduate proof of Zariski's lemma and Hilbert's nullstellensatz*, http://arxiv.org/pdf/1506.08376v1 - Goldman, Oscar (1951),
*Hilbert Rings and the Hilbert Nullstellensatz* - Grothendieck, Alexander (1957),
*Some aspects of homological algebra*(**Warning!**Only read this if you know why Grothendieck advised against reading his works!)

### Online projects[edit]

*Stacks Project*, http://stacks.math.columbia.edu/- Wikipedia