List of Abstract Algebra Topics - Basic Language

Basic Language

Algebraic structures are defined primarily as sets with operations.

  • Algebraic structure
    • Subobjects: subgroup, subring, subalgebra, submodule etc.
  • Binary operation
    • Closure of an operation
    • Associative property
    • Distributive property
    • Commutative property
  • Unary operator
    • Additive inverse, multiplicative inverse, inverse element
      • Identity element
      • Cancellation property
  • Finitary operation
    • Arity

Structure preserving maps called homomorphisms are vital in the study of algebraic objects.

  • Homomorphisms
    • Kernels and cokernels
    • Image and coimage
    • Epimorphisms and monomorphisms
      • Isomorphisms
        • Isomorphism theorems

There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra.

  • Direct sum
    • Direct limit
  • Direct product
    • Inverse limit
  • Quotient objects: quotient group, quotient ring, quotient module etc.
  • Tensor product

Advanced concepts:

  • Category theory
    • Category of groups
      • Category of abelian groups
    • Category of rings
    • Category of modules
      • Morita equivalence, Morita duality
    • Category of vector spaces
  • Homological algebra
    • Filtration (algebra)
    • Exact sequence
    • Functor
  • Zorn's lemma

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