Algebra/Ring/Direct/Definition

Ring

A set ${}R$ is called a ring if there are two binary operations (called addition and multiplication)

$+:R\times R\longrightarrow R{\text{ and }}\cdot :R\times R\longrightarrow R$

and two elements ${}0,1\in R$ that fulfill the following properties.

1. Axioms for the addition:
   1. Associative law: ${}(a+b)+c=a+(b+c)$ holds for all ${}a,b,c\in R$.
   2. Commutative law: ${}a+b=b+a$ holds for all ${}a,b\in R$.
   3. ${}0$ is the neutral element of the addition, i.e., ${}a+0=a$ holds for all ${}a\in R$.
   4. Existence of the negative: For every ${}a\in R$, there exists an element ${}b\in R$ with ${}a+b=0$.
2. Axioms of the multiplication:
   1. Associative law: ${}(a\cdot b)\cdot c=a\cdot (b\cdot c)$ holds for all ${}a,b,c\in R$.
   2. ${}1$ is the neutral element for the multiplication, i.e., ${}a\cdot 1=a$ holds for all ${}a\in R$.
3. Distributive law: ${}a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ holds for all ${}a,b,c\in R$.