List of Abstract Algebra Topics - Module Theory

Module Theory

General

• Module (mathematics)
• Bimodule
• Annihilator (ring theory)

Structure

• Submodule
  • Pure submodule
• Module homomorphism
  • Essential submodule
  • Superfluous submodule
  • Singular submodule
• Socle of a module
• Radical of a module

Constructions

• Free module
• Quotient module
• Direct sum, Direct product of modules
• Direct limit, Inverse limit
• Localization of a module
• Completion (ring theory)

Types

• Simple module, Semisimple module
• Indecomposable module
• Artinian module, Noetherian module
• Homological types:
  • Projective module
    • Projective cover
    • Swan's theorem
    • Quillen–Suslin theorem
  • Injective module
    • Injective hull
  • Flat module
    • Flat cover
• Coherent module
• Finitely-generated module
• Finitely-presented module
• Finitely related module
• Algebraically compact module
• Reflexive module

Concepts and theorems

• Composition series
  • Length of a module
• Structure theorem for finitely generated modules over a principal ideal domain
• Homological dimension
  • Projective dimension
  • Injective dimension
  • Flat dimension
  • Global dimension
  • Weak global dimension
  • Cohomological dimension
• Krull dimension
• Regular sequence (algebra), depth (algebra)
• Fitting lemma
• Schur's lemma
• Nakayama's lemma
• Krull–Schmidt theorem
• Steinitz exchange lemma
• Jordan-Holder Theorem
• Artin–Rees lemma
• Schanuel's lemma
• Morita equivalence
  • Progenerator