A quotient ring (also called a residue-class ring) is a ring that is the quotient of a ring 
by one of its two-sided ideals 
, denoted 
. For example, when the ring 
is 
(the integers) and the ideal
is 
(multiples
of 6), the quotient ring is 
.
In general, a quotient ring is a set of equivalence classes where ![[x]=[y]](/images/equations/QuotientRing/Inline8.svg)
iff 
.
When 
is a commutative ring with multiplicative
identity, the quotient ring is an integral domain iff the ideal 
is prime. A stronger condition
occurs when the quotient ring is a field, which corresponds
to when the ideal 
is maximal.
The ideals in a quotient ring 
are in a one-to-one
correspondence with ideals in 
which contain the ideal 
. In particular, the zero ideal
in 
corresponds to 
in 
. In the example above from the integers,
the ideal of even integers
contains the ideal of the multiples
of 6. In the quotient ring, the even integers correspond
to the ideal 
in 
.
