Rings

Definition

A ring is a set $R$ with two binary operations $(+,\cdot )$ such that:

1. $(R,+)$ is an abelian group
2. Multiplication is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$
3. Distributivity holds: $a\cdot (b+c)=a\cdot b+a\cdot c$ $(a+b)\cdot c=a\cdot c+b\cdot c$

A ring is commutative if $a\cdot b=b\cdot a$ for all $a,b\in R$.

Ideals

An ideal $I\subseteq R$ is an additive subgroup such that $rI\subseteq I$ and $Ir\subseteq I$ for all $r\in R$.

The quotient $R/I$ inherits a ring structure.