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

• $[a]+[b]=[a+b]$,

• $[a]\cdot[b]=[a\cdot b]$.

Here $a$ and $b$ are some elements in $R$ that represent $[a]$ and $[b]$. By construction, every element in $R/I$ has such a representative in $R$. Moreover, since $I$ is closed under addition and multiplication, one can verify that the ring structure in $R/I$ is well defined.

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 $H$.