gistfile1.md

Group

Note: * is some generic operator, not necessarily multiplication

• Closure - a in S, b in S, a * b in S
• Associativity - a + (b + c) = (a + b) + c
• Identity - for all a in S, a * i = a, where i is identity element
• Inverse - for all a in S, there exists b in S such that a * b = b * a = e where e is identity element

Ring

• Abelian Group under addition, where Albelian Group is a Group that also has commutativity
  • Commutative - a + b = b + a
• Closure under multiplication
• Multiplication is associative - (a * b) * c = a * (b * c)
• Multiplication is distributive over addition
  • Left - a * (b + c) = (a * b) + (a * c)
  • Right - (b + c) * a = (b * a) + (c * a)

Field

• Everything from Ring
• Multiplication commutativity - a * b = b * a
• Multiplicative inverses - a / b = ab^{-1}

Vector

• Distributivity of scalars - av + bu where v and u are vectors