Math Overview

Math 313 : Abstract Algebra

A Monoid is a set S with a binary operation '+', with the following properties (for A,B,C in S):
  1. Closure: A + B is in S
  2. Associative: A + ( B + C ) = ( A + B ) + C
  3. Identity: There's an I s.t. A+I = A = I+A
The set of square matrices with multiplication is a Monoid because not every square matrix has an inverse
A Group is a Monoid, with the following properties:
  1. Inverse: There's an A⁻¹ s.t. A+A⁻¹ = I = A⁻¹+A
The set of invertible matrices with multiplication is a Group (The General Linear Group)
A Commutuative Group ( Abelian Group ) is a Group G, with the following properties (for A,B in G):
  1. Commutative: A + B = B + A
The set of invertible diagonal matrices with multiplication is a Commutative Group because diagonal matrices commute
A Ring R is a Commutuative Group under '+' and Monoid under '*', with the following properties:
  1. Distributive: '*' distributes over '+'
The set of square matrices with '*' and '+' is a Ring because matrix multiplication distributes over addition
A Commutative Ring is Ring, with the following properties:
  1. Commutative: A * B = B * A
A set of similar matrices with '*' and '+' is a Commutative Ring because similar matrices commute
A Field is a Commutative Ring, with the following properties:
  1. Inverse: For A≠0, There's an A⁻¹ s.t. A*A⁻¹ = I = A⁻¹*A
The set of diagonal matrices with '*' and '+' is a Field because nonzero diagonal matrices are invertible