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.

Table of Contents[edit]

Lattices[edit]

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

Modules[edit]

  1. The lattice of submodules, Noether isomorphism theorems 100% developed
  2. Generators, chain conditions 100% developed
  3. Determinants, Cayley–Hamilton and weak Nakayama 100% developed
  4. Products and coproducts, the tensor product
  5. Localisation of modules
  6. Sequences of modules
  7. Torsion-free, flat, projective and free modules
  8. Normal and composition series
  9. Algebras

Commutative rings[edit]

  1. Basics on prime and maximal ideals and local rings 100% developed
  2. Radicals, strong Nakayama 100% developed
  3. Spectrum with Zariski topology 100% developed
  4. Jacobson rings and Jacobson spaces 100% developed
  5. Noetherian rings and spaces
  6. Primary decomposition or Lasker–Noether theory 100% developed
  7. Artinian rings
  8. Intersection and prime chains or Krull theory
  9. Valuations, (discrete) valuation rings
  10. Integral dependence

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

Sources[edit]

Books[edit]

Lecture notes[edit]

Papers[edit]

Online projects[edit]