Algebra/Ring/Direct/Definition

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.

  1. Axioms for the addition:
    1. Associative law: holds for all .
    2. Commutative law: holds for all .
    3. is the neutral element of the addition, i.e., holds for all .
    4. Existence of the negative: For every , there exists an element with .
  2. Axioms of the multiplication:
    1. Associative law: holds for all .
    2. is the neutral element for the multiplication, i.e., holds for all .
  3. Distributive law: holds for all .