# Algebra/Ring/Direct/Definition

Appearance

< Algebra

Ring

A set is called a * ring* if there are two
binary operations
(called addition and multiplication)

and two elements that fulfill the following properties.

- Axioms for the addition:
- Associative law: holds for all .
- Commutative law: holds for all .
- is the neutral element of the addition, i.e., holds for all .
- Existence of the negative: For every , there exists an element with .

- Axioms of the multiplication:
- Associative law: holds for all .
- is the neutral element for the multiplication, i.e., holds for all .

- Distributive law: holds for all .