Algebra/Ring/Direct/Definition

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

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

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

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