List of abstract algebra topics

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings.

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.