Ring

In ring theory, a ring is an algebraic sturcture consisting of a set with two binary operations. Integer equiped with addition and multiplication, polinomials, square matrices and functions are typical examples of rings.

Definition

Suppose that $\text{[math]}$ is a set and $\text{[math]}$ are two binary operations on $\text{[math]}$ .

The ordered triplet $\text{[math]}$ is a ring if it satisfies:

$\text{[math]}$ is an abelian group.
$\text{[math]}$ is a monoid.
Multiplication is distributive with respect to addition on both side:
- $\text{[math]}$ for all $\text{[math]}$ (left distributivity)
- $\text{[math]}$ for all $\text{[math]}$ (right distributivity)