Definition. Let $R$ be a ring and let $I$ be a two-sided ideal of $R$ . To define the quotient ring $R/I$ , let us first define an equivalence relation in $R$ . We say that the elements $a,b\in R$ are equivalent, written as $a\sim b$ , if and only if $a-b\in I$ . If $a$ is an element of $R$ , we denote the corresponding equivalence class by $[a]$ . Thus $[a]=[b]$ if and only if $a-b\in I$ . The quotient ring of $R$ modulo $I$ is the set $R/I=\{[a]\, |\, a\in R\}$ , with a ring structure defined as follows. If $[a],[b]$ are equivalence classes in $R/I$ , then

• $[a]+[b] = [a+b]$ ,
• $[a]\cdot [b]=[a\cdot b]$ .

A common notation is $a+I=[a]$ which is consistent with the notion of classes $[a]=aH\in G/H$ for a group $G$ and a normal subgroup $H$ .

Properties

1. If $R$ is commutative, then $R/I$ is commutative.
2. The mapping $R\to R/I$ , $a\mapsto [a]$ is a homomorphism, and is called the natural homomorphism.

Examples

1. For a ring $R$ , we have $R/R=\{[0]\}$ and $R/\{0\}=R$ .
2. Let $R=\mathbb{Z}$ , and let $I=2\mathbb{Z}$ be the set of even numbers. Then $R/I$ contains only two classes; one for even numbers, and one for odd numbers. Actually this quotient ring is a field. It is the only field with two elements (up to isomorphy) and is also denoted by $\mathbb{F}_2$ .
3. One way to construct complex numbers is to consider the field $\mathbb{R}[T]/(T^2+1)$ . This field can viewed as the set of all polynomials of degree $1$ with normal addition and $(a+bT)(c+dT)=ac-bd+(ad+bc)T$ , which is like complex multiplication.