Definition of a Ring

A ring is a set $R$ with two operations, usually denoted by $+$ and $\cdot$, satisfying:

• $R$ with $+$ is an abelian group.
• $R$ is closed with respect $\cdot$, i.e. $a\cdot b\in R,\forall a,b\in R$.
• $\cdot$ is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
• For all $a,b,c\in R$, the distributive laws hold:
  • $a\cdot (b+c)=a\cdot b+a\cdot c$.
  • $(a+b)\cdot c=a\cdot c+b\cdot c$.

Notation: Suppose a set $R$. If $R$ is a ring with operations $⨁$ and $⨀$, we write $(R,⨁,⨀)$. If the multiplicative operator is omitted, the group $(R,+)$ is an additive group.