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