Ring Theory
Table of Contents
- 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
1. Commutative Ring
- A ring where the multiplication is commutative is called Commutative Ring or Commutative Algebra.