Ring Theory

Ring theory is the study of rings, an algebraic structure where two operations addition and multiplication are defined.

• A set \(R\) equiped with two operators addition and multiplication such that (they satisfy the usual property in integers (commutativity in multiplication is not required)):
  • \(R\) is Monoid under multiplication (i.e. associative, has identity)
  • \(R\) is Abelian Group under addition (i.e. associative, has identity, invertible, commutative)
  • Multiplication is distributive over addition

• In an nontrivial ring, the additive identity and multiplicative identity must be different.

  If 0 is additive identity and 1 is multiplicative identity, then

  \begin{equation*} r = r \times 1 = r \times 0 = 0 \end{equation*}

  i.e. the Ring is trivial. \(R = {0}\)

• Examples:
  • Integers
  • Complex Numbers
  • Polynomials
  • Square Matrices